Allies of Jeremy Corbyn have publicly urged the Labour leader to commit to retaining the free movement of workers between Britain and the European Union – as the party’s Brexit stance comes under increasing scrutiny.

Senior figures on the left of the party have set up a new grouping, the Labour Campaign for Free Movement, and issued an ultimatum to the leadership that says Labour must be “the party of all working people, regardless of where they were born”.

Labour MPs and MEPs including Clive Lewis and David Lammy are backing the new group, as well as left-wing trade union chiefs and members of the party’s ruling National Executive Committee.

Union general secretaries endorsing the new campaign include Manuel Cortes, of the Transport Salaried Staffs’ Association, Sally Hunt of the University and College Union, and Ronnie Draper of the Bakers Food and Allied Workers Union – figures usually considered allies of Mr Corbyn. Journalists and campaigners such as Owen Jones, Dawn Foster and Neal Lawson have also signed a statement in support of the group.

The group says that Labour, like the UK in general, is “at a crossroads” in its relationship with the world and that the party should respond with “clarity, humanity, and solidarity”.

The latest intervention is significant because it mostly comes from the Corbyn-allied left of the Labour Party – in contrast to calls to remain in the single market, which have mostly, though not entirely, been championed by Labour centrists.

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However, the issue of the single market and free movement are broadly linked, with single market access effectively contingent on free movement.

Labour’s manifesto was vague on the issue of Brexit, stating that freedom of movement “will end” when Britain leaves the EU as if it were a matter of fact. However, many countries remain outside the EU but have free movement with the bloc, such as Norway, Switzerland, and Iceland.

The party has walked a delicate balancing act to neutralise immigration as an issue.

Theresa May has said she will end free movement but not given any concrete suggestions of what system might replace it.

“We fought the last general election arguing against such scapegoating, and celebrating the contributions of migrants to our society. That tone must now translate into policy,” the joint statement says.

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“Migrants are not to blame for falling wages, insecurity, bad housing and overstretched public services. These are the product of decades of underinvestment, deregulation, privatisation, and the harshest anti-union laws in Europe. On the contrary, migrant workers have been on the front line of fighting for better pay and working conditions. Labour is the party of all working people – regardless of where they were born.

“A system of free movement is the best way to protect and advance the interests of all workers, by giving everyone the right to work legally, join a union and stand up to their boss without fear of deportation or destitution.

Clive Lewis is among MPs to have backed the new campaign (PA)

“Curtailing those rights, or limiting migrants’ access to public services and benefits, will make it easier for unscrupulous employers to hyper-exploit migrant labour, which in turn undermines the rights and conditions of all workers.”

Mr Corbyn said last month that he would not allow the “wholesale importation” of EU workers to undercut wages – a sharp contrast to the tone he took last year when he said he did not think there were too many EU migrants in Britain.

As with the Government, it is not clear what Labour’s actual post-Brexit policy on free movement would be – though barring an early general election the party is unlikely to actually be directly involved in creating a new system.

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Michael Chessum, an organiser for the Labour Campaign for Free Movement, said: “Labour’s immigration stance has for far too long been dominated by pandering to the idea that immigration is to blame for a fall in living standards. This isn’t just factually wrong, it’s also self-defeating – because we need a narrative that is clear and honest about the fact that neoliberalism and exploitation are the real problem.