I’m in favour of immigration, and I’m not a socialist.
But there are several distinct left-wing arguments against. I mostly disagree with the below but I can’t see how any of them can be called far right (except arguably the final one, on a horseshoe principle).
The most convincing, to my mind, is the idea that permissive migration and a strong welfare state are incompatible. Both on financial grounds and because migration tends to undermine the solidarity needed to win support for welfare. That seems to hold up empirically pretty well.
Another is that fairness, in a democratic socialist system, is owed first to the working classes of that demos. That immigration tends to undermine and compete with the working class, for the benefit of the capitalist classes. This view might limit numbers, or types of occupation a migrant can take up.
A third is that it leads to exploitation of migrants in the destination country. A version of the above, but out of paternalistic concern for the migrants. This view probably approves of many forms of migration, but still imposes limits and controls (eg on minimum wages a migrant could take). The migrant is taken not to be able to choose what is best for them.
A fourth is that migration allows the talented and mobile to opt out of their countries to seek their betterment elsewhere. That it brain-drains and hollows out societies, especially poor ones, holding back national development that benefits all, for the personal enrichment of those who leave and their families. That it’s fundamentally anti-communitarian.
A fifth is about cultural displacement (a sort of gentrification argument). I don’t think I’ve heard this from the left about the UK, but certainly about elsewhere. Usually concerns that migrants from a richer or more dominant culture (eg, Westerners, Chinese diasporas, Indians) are driving out or despoiling a local culture taken to be more worthy.
A sixth is about high-skill/rich migration. The anti-elite arguments (sometimes tied in with conspirisicm or antisemitism) about global capitalists and globalism.
And finally, though it’s not really an argument, the historical fact that several explicitly socialist/ communist governments in the 20th century have had extremely strict border controls, near-prohibiting people either entering or leaving.