It would be nice to try to catalogue a series of events that benefitted immigrants, and then gauge any resulting swings in voter sentiment among Hispanics. Jon, knowing you I imagine that's what you're looking for...numbers.
But aside from that, I agree--it does seem intuitive/obvious that loosening immigration restrictions is likely to benefit (politically) the person doing it, at least, in the limited form of more political support from the demographics that benefit from the loosened restrictions.
The best the Republicans can do here is the same thing they always have to do -- appeal to the character and pride of people who want rules to be followed, and work/opportunities to be rewarded, rather than getting freebies. It's the classic conservative vs. liberal dilemma. I love the fact that I might get a $1,400 or $2,800 check from Joe Biden, but my character stops me from deciding how to vote based on getting a check from someone. Not everyone thinks like that. And I do try to understand that the more materially desperate one is, the less one can afford the fancy distinction I'm making, and the more understandable their choice is.
Maybe Republicans can compete somewhat with liberals' border-freebies by offering to make some very carefully vetted, wise investments in South America rather than just letting people in here. Because if there is one thing I've learned from marrying into a Hispanic family (a truly non-American fully Hispanic family), it's that some of us grew up with the MISTAKEN impression that everyone is just dying to get to the USA. The truth is more along the lines of, a lot of people want to come here, but only because of $--IF they had the option of doing just a bit better in their homeland, they would greatly prefer to stay there and do a little better. They don't "like the USA" for anything other than the dollars it gives them to help them get ahead (which I totally understand--there is a LOT to like about Hispanic countries & cultures, over & above the USA).