Again, becoming pro-immigrant will probably not help Republicans very much
I have said this before, but it is worth repeating given recent discussions among Republicans about immigration policy: becoming pro-immigration, even ostantatiously pro-immigration, even more pro-immigration than the Democrats, is probably not a cure-all for the Republican Party's demographic problems.
I made this point in Real Clear Policy back in 2013, and I stand by that position. See the original piece for my entire argument.
Here is my basic point: immigrants, and the children of immigrants, are (on average and controlling for all other variables) to the Left of the rest of the American public on a whole range of other issues.
Let's say that immigrants actually are single-issue voters, and currently never vote Republican because they view the Democrats as better on immigration. What happens if immigration is off the table because the GOP moves Left on the issue? Presumably, immigrants will start voting on other issues. Do they support the Republican platform on all these other issues more often than they support the Democratic platform? No. Maybe that will change in the future, but for now, immigrants are, on average, pretty liberal.
I should note that this is not an argument for or against a particular immigration policy. Presumably a policy as important as immigration should be determined by considerations other than mere partisan gain or loss. That being said, the argument that Republicans will enjoy a huge electoral windfall if they become more enthusiastic about immigration and pursue policies that increase the number of immigrants to the U.S. is clearly wrong.
I suspect many of the people telling Republicans that they need to get on board with liberalizing immigration reforms are well aware of this and are concern trolling the GOP. One wonders what the response will be when a Republican congress eventually backs a pathway to citizenship and their numbers do not subsequently go up among immigrant groups. My guess is that the Republican grassroots will be pretty ticked.