What is the left wing policy on immigration? Historically free movement of people has been a right-wing policy akin to the free market. (See America for most of its history) For most of the last century left wing societies in Europe didn't let you move abroad - many required government permission to move cities even.
I don't think it's a left/right thing. It's about finding a solution that works and at the very least a solution that doesn't sub-contract out the process to global human trafficking gangs to enrichen themselves by providing the service.
I agree though that many Labour voters are not particularly "left wing". And nor should they be. Most people just want a decent society and dont really care which ideological branch of this social science provides it. It's as relevant as religious doctrine to the majority of people.
And thank f**k for that. Imagine how awful the world would be if everyone spoke like we do on here