No one is going to believe the GOP if they suddenly start supporting immigration reform. Probably not even as much as people believe democrats that support the keystone pipeline - or some democrats that support gun control.
The problem also is primaries. My understanding is that (especially for republicans) - the more extreme sides seem to vote more in primaries. Therefore tea party candidates have a better shot - as you get more extreme voters out to the polls. Candidates that espouse more moderate positions don’t seem to do so well - and can’t even win a primary often in order to be able to run in a general election (where they might win).
There are exceptions to this - as John McCain won in part as republicans sometimes didn’t like him, but thought he had the best chance of beating Hillary Clinton (who many thought was going to win).
The republicans are IMHO screwed as they can’t afford to upset their base - which is dying off - and can’t attract new members - as the stuff they need to attract them pisses off the old members.
I’m sure they will try some way around this - and I haven’t been paying much attention for the last couple years - so maybe they already are. They are politically savvy, but will need to come up with something even better than the southern strategy.