> So far in the US, most labor unions have generally been fairly left-wing, including being pro-immigration.
The line up against which leftist ideology fails to cohere with capitalism. Labor unions are a leftist concept but their goal is to uphold worker's rights against Capital (corporations or whoever can swing money around, like lobbyists), a pretty stark Marxist concept. However another thing Marx pointed out is that capitalism is virulent - it will industrialize a country, lift out the working class to middle class and create petit bourgeois, and then move elsewhere to seek new forms of cheap labor. What does your local union do when that happens?
I suppose this is why it was always "Workers of the World" not "Workers of Local 816," but so far as I can tell there's not much organization across international worker's groups outside of explicitly communist groups that are basically clubs, not actual unions with any power.