The four contenders clash over immigration, benefits cuts, “making excuses” for Vladimir Putin and the shadow of the Iraq war.
All four candidates said more action is needed to tackle the unfolding migration crisis at Europe’s borders.
Mr Burnham said: “As this crisis unfolds you cannot let Greece particularly shoulder the burden, given all the problems that the Greece government are facing. There has to be a European approach here where everybody agrees to play their part.”
Following remarks this morning about Britain being able to absorb 10,000 asylum seekers, Ms Cooper said she wanted local authorities to volunteer to take people in.
We have to act. We cannot turn our backs. Yvette Cooper
She said: “Since I called for action this morning you’ve had councillors and people coming forward saying they want to do something to help.
“In the end the final number would depend on what the communities came forward to do. That’s the best way. I think when you ask people, they actually want to come forward.
She added: “This is a humanitarian crisis, the greatest we have seen since the Second World War. We have to act. We cannot turn our backs.”
Liz Kendall said: ” How you come up with any decent solution is making sure you’ve got the resources to cope and the willingness to put a scheme in place and that you negotiate with your neighbours.
“But there’s more we also need to be doing to make sure that when people arrive in places like Greece that they are properly assessed. We aren’t putting the money in there either.”
Mr Corbyn said: “In these dreadful scenes today in Hungary where there’s thousands of refugees trying to get on trains and not being allowed to.
“They’re going to be pushed back over the border into Serbia according to the Hungarian government.
It’s a pretty disgraceful set of situations where you’ve got desperate people being pushed around. The British government should intervene along with others.
All of the candidates reserved judgment on the prospect of former Prime Minister Tony Blair going on trial for war crimes over the invasion of Iraq while they lead the Labour party.
Mr Corbyn said: “If there’s a (criminal) charge against anyone on this then they must face it.”
Ms Cooper responded: “I don’t think we should just rush to judgement when we haven’t had the Chilcot report out. Making these sweeping political statements I think is not helpful.”
Andy Burnham said: “This was an intense period in our country’s history when decisions were taken in the aftermath of the attack on the World Trade Center, Afghanistan.
The next Labour leader has got to lead our party out of the shadow of Iraq. Liz Kendall
“It’s easy for people to sit here today and say it was all so easy and that people knowingly misled people. I don’t believe that to be the case, having lived through that period in time. There was a widespread view that Saddam had weapons of mass destruction.”
Ms Kendall said: “The next Labour leader has got to lead our party out of the shadow of Iraq.
“I think it has prevented us from having a sensible debate about Britain’s role in the world and a future foreign policy at precisely the time when David Cameron has overseen a quite diminishing of Britain’s role in the world.”
Andy Burnham accused Jeremy Corbyn of “making excuses” for Russia’s actions in Crimea and eastern Ukraine after Mr Corbyn criticised “Nato’s excessive and obsessive expansion”.
The left-winger has questioned Britain’s continuing membership of Nato and wrote in an article last year: “On Ukraine, I would not condone Russian behaviour or expansion. But it is not unprovoked, and the right of people to seek a federal structure or independence should not be denied.”
In today’s Channel 4 News debate, he said: “I’m not a supporter of Putin, Russian expansion or anybody else’s expansion. I thought that Nato’s excessive and obsessive expansion since 1990 has been a problem.”
It sounds like you’re making excuses for Putin there. Andy Burnham
Mr Burnham said: “It sounds like you’re making excuses for Putin there… you’re saying he was prompted and provoked into doing what he did in the Ukraine.
“We’ve seen a breach here in terms of the sovereignty of a country and we shouldn’t make excuses for that.”
Mr Burnham added: “Internationalism should be our approach as a party. We shouldn’t be talking about leaving Nato.”
Yvette Cooper said the evidence was that Mr Putin “doesn’t just back off in response to soft power”.
“The evidence, sadly, on Putin is that in the end it needs a firm security response in order for him to back off,” she added.
Mr Corbyn came out against restricting migrant workers’ right to claim welfare, and said Labour should oppose the Conservatives’ benefit reform bill in its entirety, including the decision to cut the household benefits cap, saying it would lead to “social cleansing of towns and cities all over Britain”.
We’ve got to oppose this bill in its totality for what it’s trying to do. Jeremy Corbyn
He added: “We’ve got to oppose this bill in its totality for what it’s trying to do – it’s trying to hurt the most vulnerable and weakest in our society.”
Mr Burnham said benefits cuts was “where Labour needs to find the courage of its convictions”.
Asked whether she would support cuts to welfare payments for migrants, Ms Kendall said: “Without doubt I believe in the contributory principle that you should pay in before you take out.”
Mr Corbyn criticised the last Labour government’s use of PFI contracts to fund capital investment, but his own plans for “people’s quantitative easing” – a form of QE where money is created by the Bank of England and used to fund infrastructure rather than going to the banks – came under attack.
Your proposal is just like PFI is steroids. Yvette Cooper
Ms Cooper said: “The economy is growing. The growth is late, but it is growing If you want to just print money when the economy is growing that’s just going to be put up inflation. It’s not real.
“Your proposal is just like PFI is steroids. This is not a way in which you can genuinely fund infrastructure.
“It’s just not responsible and it will be ripped apart by economists.”