They already did, and it was forcasted that all the swing states would be purple in this election instead of red.
Problem is, Democrats moving to a Red Area are facing less population density and so larger estates, less taxes and find pretty quickly they don't have to worry about crime as much, and that the racism whole not non-existent isn't nearly as bad as they were lead on to believe, and is infact faw worst in blue states.
So many decided to stay culturally liberal but switch to more conservative voting behavior.
You have a risk of thw negative behavior as in the OP when it is a mass movement, like when Californian businesses moved to Texas in places such as Austin. The owners wanted conservative government, but their employees who made the move, not so much. They came as a tribe of expats in a foreign, alien land and preserved their politics. But that is eroding because when you drive out of Austin you don't suddenly have the issues of backwards decline. The state level isn't exactly thrilled with your policies, and ymmany of your neighbors aren't into what you are into. So you adapt. You widen your social net, give up a bit on some ideas and adopt new ones. This process doesn't stop till they turn red.