It's true that immigration can have downsides for workers if a) you are in a neoliberal capitalist shithole, and b) if you have zero solidarity between native and immigrant workers such that immigrants are allowed to be abused and super-exploited as a consequence of their more vulnerable status.

The way to fight things like downwards wage pressure resulting from immigration is through worker solidarity, unionization and other forms of collective action and organization. And yes this is possible to do with people of immigrant background too; after all, in today's day and age, with the technology available, language should be no barrier to organizing together. The only real barrier is racism.

Standing up for the rights of migrants and fighting for the rights of native workers, these are deeply interlinked struggles and the success of one will always positively affect the other. Throwing migrants under the bus in hopes of bettering native labor's position on the other hand is always going to end up having the opposite effect: it empowers capital while it atomizes and fragments the working class.

That's not to say that massive migration is necessarily a good thing. While migration can have very beneficial effects on the receiver country it is often detrimental to the countries of origin which experience brain drain and loss of the most highly trained labor. We also cannot forget that highly unbalanced migration is always caused by something being very wrong and pushing people to feel forced to leave their native lands, something which most people would not freely choose to do under normal circumstances.

Whether it's war, economic deprivation or ecological disasters that are causing this migration, those causes (most of which were brought about by the same neoliberal imperialist global system that also seeks to profit from turning native and migrant labor against each other in order to better exploit both) are what we should be looking at and helping those countries to tackle. We should not merely be shutting people out because that does not fix the problem, it only causes more suffering.

We are Marxists, we have to take a holistic view on these issues and not deal with them in the simplistic, un-dialectical way that a liberal or a fascist would.