Well, then the question is are these Republicans moving to more friendly turf or Democrats chasing the jobs?
In regards to Texas, it is definitely normal people chasing jobs. And since they come from liberal places, they will also tend to be more liberal than Texans. And I think this will be especially true of social issues, as opposed to things like opposing a state income tax which should be popular with the new migrants as well.
Then you have a place like Arizona or South Carolina which is mostly retirees from the big states moving there. And these people tend to be quite Republican and I doubt will change the local political culture much.