Transport Secretary Chris Grayling described departures from the cabinet in recent
months as "unwanted noise".
Amber Rudd is the fourth person to leave the cabinet in the last six months - following Sir Michael Fallon, Priti Patel and Damian Green.
Mr Grayling told BBC Radio 4's Today programme: "We have had some unwanted noise in the last few months, some
unwanted loss of parliamentary colleagues and Cabinet colleagues.
"None of that are things we would have wanted to happen.
"They happen to all
governments to some degree at some point or another."
Diane Abbott: PM has questions to answer
Video content
Video caption: Labour's Diane Abbott says Theresa May needs to be asked more about her role on Windrush.Labour's Diane Abbott says Theresa May needs to be asked more about her role on Windrush.
Theresa May still has questions to answer on the "Windrush fiasco", says Labour's Diane Abbott.
Speaking to the BBC's Today programme, the shadow home secretary said Amber Rudd was right to resign as "somebody had to take responsibility".
Conservatives 'inherited hostile environment' at Home Office
BBCCopyright: BBC
A former immigration minister has said the Conservatives "inherited" a "hostile environment" at the Home Office but said that Ms Rudd made the right decision in standing down.
Damian Green, who resigned from the cabinet last year, worked alongside the then home secretary Theresa May in 2010 - during the destruction of landing cards of migrants at the Home Office, which has contributed to the Windrush scandal.
He said he did not know about the destruction of those documents.
Mr Green praised Amber Rudd as an "excellent home secretary" for her work on counter-terrorism and tackling modern slavery.
Favourites to replace Rudd
PACopyright: PA
The prime minister is set to announce Amber Rudd's replacement this morning.
Theresa May will be looking for a home secretary who can help balance of power between Leave and Remain-backing cabinet members.
So who are among the runners and riders?
Sajid Javid: The communities secretary is currently the bookies' favourite. He is seen as a uniting figure and said of the Windrush scandal: "It could have been me, my mum or my
dad." A Remain campaigner in the EU referendum, his appointment would balance the scales of Brexiters and Remainers in the highest positions in the cabinet.
Michael Gove: The environment secretary, a leading Brexiter, has impressed many with his latest portfolio - particularly with the single-use plastic and microbead campaigns. But it follows the humiliation of a botched leadership bid in 2016 and Theresa May firing him as justice secretary when she became PM.
Jeremy Hunt: The health secretary is one of the great survivors in cabinet. He has clung on to his job despite the junior doctors strikes and the NHS winter crisis. But he is also under investigation over breaches of parliamentary conduct over the purchase of luxury flats, which he has called an "honest mistake". He campaigned to Remain, but has since said he has changed his mind.
Karen Bradley: The newly appointed Northern Ireland secretary has risen up the ranks in cabinet recently, but Mrs May might not want to take her away from her most recent brief. She campaigned for Remain.
James Brokenshire: The former Northern Ireland secretary had to step down from cabinet due to ill health after being diagnosed with the early stages of lung cancer. But the former immigration secretary is back in Parliament now. He voted to Remain.
More headaches for May after fifth resignation in six months
BBCCopyright: BBC
Theresa May has struggled to keep her cabinet in one piece after three key figures were forced to quit over the past six months, with another minister resigning for health reasons. Here are the other four resignations:
Sir Michael Fallon: The former defence secretary was forced to leave his post after accusations he had acted inappropriately towards several female journalists. He resigned in November admitting his "previous conduct" had "fallen below" what was acceptable.
Priti Patel: The former international development secretary was forced to fly back from Kenya after it was revealed she had meetings with Israeli officials, without informing the Foreign Office or Downing Street. She was sacked in November.
Damian Green: The former first secretary of state was a key ally of Theresa May and considered to be her second in command. But he was sacked in December after an inquiry found he had breached the ministerial code. It followed claims that pornography was found on his office computer in 2008 and allegations of inappropriate behaviour towards a journalist and Tory activist.
James Brokenshire: Unlike the other resignations, the former Northern Ireland secretary had not faced calls to resign. After being diagnosed with the early stages of lung cancer, he left his cabinet position on medical grounds in January.
PM 'should now face questions'
BBCCopyright: BBC
The shadow home secretary Diane Abbott has said the prime minister should now face questions in Parliament about illegal immigrant removal targets.
Amber Rudd admitted she "inadvertently misled" MP about her knowledge of the removal targets.
Ms Abbott added she wants Mrs May to discuss the so-called "hostile environment" at the Home Office, which she says is the prime minister's responsibility as she was previously home secretary.
She added Ms Rudd was right to resign over the Windrush scandal because "someone had to take responsibility".
Cabinet reaction to Rudd's resignation
Here's a look at how Amber Rudd's former cabinet colleagues are responding to her departure:
Amber Rudd's resignation over the question of whether she knew about targets to remove illegal immigrants has shown once again that being home secretary remains one of the most high-risk jobs in government.
Before the full extent of the Windrush controversy became clear, shining an unflattering light on the workings and culture of her department and Ms Rudd's own command of her brief, the 54-year-old had been regarded as one of the Conservatives' rising stars.
She was promoted rapidly under David Cameron and then replacing Theresa May as home secretary in 2016.
Her first year in the Home Office went smoothly enough, although dealing with the succession of terror attacks that took place during 2017 was a harrowing experience.
But in the face of daily revelations about the plight of Windrush migrants and their families, Ms Rudd appeared slow to respond and, despite numerous apologies, to appreciate the scale and significance of the crisis.
Political commentators are considering how Amber Rudd's departure will affect Prime Minister Theresa May, who was home secretary for six years before moving to Downing Street.
Spectator political editor James Forsyth said Ms Rudd's departure "removes the firebreak between May and the whole Windrush scandal".
Newspaper headlines: Updated Telegraph on Rudd resignation
The Daily Telegraph has updated its front page story on Amber Rudd's resignation after the former home secretary admitted "inadvertently" misleading Parliament on immigration figures.
Boris Johnson and Amber Rudd may not see eye-to-eye over Brexit but the foreign secretary has said he is "very sad" to lose his colleague from the cabinet.
Labour leader Jeremy Corbyn is yet to react publicly to the home secretary's resignation but shadow chancellor John McDonnell has said the development is evidence of a government "decomposing".
For readers just waking up to the news about Amber Rudd's resignation, here is the story so far:
After a weekend of intense political pressure stemming from her handling of the Windrush scandal, the home secretary resigned from her post on Sunday night.
In a letter to the prime minister, Ms Rudd admitted she had "inadvertently misled" MPs over migrant deportation targets and, as such, would step down from her role.
Shadow home secretary Diane Abbott said Ms Rudd had "done the right thing" while many Conservative MPs offered support.
Environment Secretary Michael Gove - who is among the candidates for the vacant position - tweeted that it was "sad" that the government had lost "a huge asset".
Video content
Video caption: Amber Rudd: Why has the home secretary resigned?Amber Rudd: Why has the home secretary resigned?
When will the new home secretary be announced?
The prime minister is expected to announce Ms Rudd's successor early on Monday morning.
The Guardian's Amelia Gentleman - who broke the Windrush story that has now led to the home secretary's resignation - says Windrush victims have told her they are surprised by Ms Rudd's resignation.
Live Reporting
Paul Gribben, Francesca Gillett and Claire Heald
All times stated are UK
Pic: Protesters outside Home Office
Gove 'very sad' over Rudd
Michael Gove, the environment secretary, is among the contenders to replace Amber Rudd as home secretary.
He told reporters this morning: "I'm very sad that Amber has had to leave the government and I very much hope she'll be back... soon."
'Delicate balancing act for May'
Our political editor tweets
Laura Kuenssberg
BBC political editor
Resignations 'unwanted noise'
Transport Secretary Chris Grayling described departures from the cabinet in recent months as "unwanted noise".
Amber Rudd is the fourth person to leave the cabinet in the last six months - following Sir Michael Fallon, Priti Patel and Damian Green.
Mr Grayling told BBC Radio 4's Today programme: "We have had some unwanted noise in the last few months, some unwanted loss of parliamentary colleagues and Cabinet colleagues.
"None of that are things we would have wanted to happen.
"They happen to all governments to some degree at some point or another."
Diane Abbott: PM has questions to answer
Video content
Theresa May still has questions to answer on the "Windrush fiasco", says Labour's Diane Abbott.
Speaking to the BBC's Today programme, the shadow home secretary said Amber Rudd was right to resign as "somebody had to take responsibility".
VIDEO: Why has Amber Rudd resigned?
Conservatives 'inherited hostile environment' at Home Office
A former immigration minister has said the Conservatives "inherited" a "hostile environment" at the Home Office but said that Ms Rudd made the right decision in standing down.
Damian Green, who resigned from the cabinet last year, worked alongside the then home secretary Theresa May in 2010 - during the destruction of landing cards of migrants at the Home Office, which has contributed to the Windrush scandal.
He said he did not know about the destruction of those documents.
Mr Green praised Amber Rudd as an "excellent home secretary" for her work on counter-terrorism and tackling modern slavery.
Favourites to replace Rudd
The prime minister is set to announce Amber Rudd's replacement this morning.
Theresa May will be looking for a home secretary who can help balance of power between Leave and Remain-backing cabinet members.
So who are among the runners and riders?
More headaches for May after fifth resignation in six months
Theresa May has struggled to keep her cabinet in one piece after three key figures were forced to quit over the past six months, with another minister resigning for health reasons. Here are the other four resignations:
PM 'should now face questions'
The shadow home secretary Diane Abbott has said the prime minister should now face questions in Parliament about illegal immigrant removal targets.
Amber Rudd admitted she "inadvertently misled" MP about her knowledge of the removal targets.
Ms Abbott added she wants Mrs May to discuss the so-called "hostile environment" at the Home Office, which she says is the prime minister's responsibility as she was previously home secretary.
She added Ms Rudd was right to resign over the Windrush scandal because "someone had to take responsibility".
Cabinet reaction to Rudd's resignation
Here's a look at how Amber Rudd's former cabinet colleagues are responding to her departure:
What went wrong for the former home secretary?
Gavin Stamp
BBC political reporter
Amber Rudd's resignation over the question of whether she knew about targets to remove illegal immigrants has shown once again that being home secretary remains one of the most high-risk jobs in government.
Before the full extent of the Windrush controversy became clear, shining an unflattering light on the workings and culture of her department and Ms Rudd's own command of her brief, the 54-year-old had been regarded as one of the Conservatives' rising stars.
She was promoted rapidly under David Cameron and then replacing Theresa May as home secretary in 2016.
Her first year in the Home Office went smoothly enough, although dealing with the succession of terror attacks that took place during 2017 was a harrowing experience.
But in the face of daily revelations about the plight of Windrush migrants and their families, Ms Rudd appeared slow to respond and, despite numerous apologies, to appreciate the scale and significance of the crisis.
Read the full article here.
Does Rudd's departure leave May vulnerable?
Political commentators are considering how Amber Rudd's departure will affect Prime Minister Theresa May, who was home secretary for six years before moving to Downing Street.
Spectator political editor James Forsyth said Ms Rudd's departure "removes the firebreak between May and the whole Windrush scandal".
Newspaper headlines: Updated Telegraph on Rudd resignation
The Daily Telegraph has updated its front page story on Amber Rudd's resignation after the former home secretary admitted "inadvertently" misleading Parliament on immigration figures.
What to read on your commute
For those looking for more detail and analysis on Amber Rudd's resignation, here is some morning reading for your commute:
Johnson 'really sad' to lose Rudd from cabinet
Boris Johnson and Amber Rudd may not see eye-to-eye over Brexit but the foreign secretary has said he is "very sad" to lose his colleague from the cabinet.
Tories in 'terminal disarray'
Labour leader Jeremy Corbyn is yet to react publicly to the home secretary's resignation but shadow chancellor John McDonnell has said the development is evidence of a government "decomposing".
What you missed overnight
For readers just waking up to the news about Amber Rudd's resignation, here is the story so far:
After a weekend of intense political pressure stemming from her handling of the Windrush scandal, the home secretary resigned from her post on Sunday night.
In a letter to the prime minister, Ms Rudd admitted she had "inadvertently misled" MPs over migrant deportation targets and, as such, would step down from her role.
Theresa May said she was "very sorry" to lose her colleague from the cabinet - and now faces the delicate task of finding a replacement.
Shadow home secretary Diane Abbott said Ms Rudd had "done the right thing" while many Conservative MPs offered support.
Environment Secretary Michael Gove - who is among the candidates for the vacant position - tweeted that it was "sad" that the government had lost "a huge asset".
Video content
When will the new home secretary be announced?
The prime minister is expected to announce Ms Rudd's successor early on Monday morning.
However, as BBC political editor Laura Kuenssberg warns, Mrs May will be conscious of upsetting the delicate balance within her cabinet.
Windrush victims 'surprised by Rudd resignation'
The Guardian's Amelia Gentleman - who broke the Windrush story that has now led to the home secretary's resignation - says Windrush victims have told her they are surprised by Ms Rudd's resignation.