A popular right-wing complaint about immigration is that the left doesn’t want immigrants to integrate. I don’t think this is true. Both the left and the right want new arrivals to their culture to share their values, they just aren’t on the same page about which values.
I’m not actually sure where the misunderstanding comes from. You see a lot of conservatives making claims like ‘liberals think that beating your spouse should respected if it’s part of your (minority) culture’ and I’ve yet to run across a single liberal saying such a thing. Maybe they mean something more like ‘liberals think that diversity is great and more important than actual correctness, so when they embrace diverse viewpoints they end up embracing abhorrent ones’. Only elsewhere conservatives (reasonably) make the complaint that the left is not remotely concerned with viewpoint diversity, just with the ‘different sets of oppressions’ kind, and accordingly is hostile to underrepresented perspectives like the evangelical one or the Appalachian ex-coal-miner one. Do they think that the left is more tolerant of ‘sometimes beating your spouse is okay’? or ‘being gay is okay unless you’re Muslim’?
I think that’s part of it; the left is definitely less viscerally mad about Muslim oppression of LGBT people, mostly because how mad we are is related to our own experiences and most of us have dealt with intolerant Christians in positions of structural power and not with intolerant Muslims in same, and the right is right here passing laws right now while ISIS is very far away and everyone agrees they’re terrible so they’re hard to get too worked up about.
But maybe most of the difference is in where the attribution lies for the bad beliefs. The left will tend to say that the beating spouses is patriarchy and the oppression of LGBT people is homophobia and transphobia, and the conservative will say they are both Islam. Then the liberal interprets the conservative as trying to dodge responsibility for the much more rampant sexism and homophobia and transphobia of the Christian right, and the conservative interprets the liberal insistence that it’s nothing to do with the actual religion which the people in question would cite as the basis for their beliefs as proof they don’t actually care about fixing those things, or why would they ignore the obvious cause staring them in the face?
And the difference lies in which things they want erased by integration: conservatives want immigrants to have conservative values, like learning English and being a reliable voting bloc against abortion and not relying on handouts and not wearing hijabs and eating American food, and liberals want immigrants to have liberal values like diversity of foods and clothing styles and languages and supporting Palestine and hating the right.
But there’s still a problem. As far as I can tell, Hispanic immigrants are that ideal conservative immigrant: Christian, socially conservative, no hijabs, integrating fast, and so forth. And conservatives can’t stand them either. I’m not sure if that means that integration is a red herring and not actually what anyone cares about, or if it’s another weird artifact of the way the American immigration debate borrows talking points from the European one despite totally incomparable situations or what. But I think firm commitment to immediate total integration would do very little about conservative opposition to immigration.
I don’t think it’s about values. I’ve never seen a conservative worked up about immigration from a white country (for a reasonable definition of white). This despite the fact that, say, a resident of Berlin considering moving to America quite possibly has more alien values on average than a resident of “random town, South America”, with a reference point of someone who clings to their guns and religion. More carefully, I think it’s about ethnicity, not race or skin color.