Considering the history of labor unions and their support for the banning of Asian citizenship in the US, I think the right frame is that migration support / opposition has something to do with other things but not entirely. There are only a few ways to come to the US and the employment-based approach is probably the most widely used by unconnected foreigners seeking to live in the US and that pathway is intended to be abolished by Senator Bernie Sanders with no replacement planned. I think it's hard to describe that senator as particularly right-wing.

Likewise, it was a labour union in Georgia that lobbied to get Koreans deported from the US. It should be unsurprising that those who hold to the Lump of Labor economic school should oppose immigration and that's quite popular among the left-wing.