This has concerned and intrigued me for years. Conventional wisdom says they'll despoil their new home with liberal voting tendencies. Such conventions often are wrong. Hispanics oppose illegal immigration, no walls, and abortion--usually because of strong Catholic upbringings. Blexit and Walk-Away movements indicate blacks are slowly moving away from liberal D ideology--again often for religious reasons.
Should a liberal voter replant, he'll likely be in a more conservative area--perhaps a dramatically more conservative area into which they naturally want to be accepted. They'll be exposed to ideas which were never expressed in NY or San Fran, or Chicago or Detroit. They may resist but won't be able to miss the greater freedoms, fewer restrictions in Florida or Texas or in 'flyover country'.
Some will resist for a while, but drastic flips of all red states to blue will become another dispelled piece of conventional wisdom, imho.
Then the housing boom hit our area, people moved from Chicago, the democrats accumulated to the point where some group that called itself "McHenry County Democrats" had enough people to have an annual picnic, and the next thing you know the Chicago suburbs and collar counties started electing Democrats.
You may be correct about people moving from a Blue area turning Red in their voting habits, but please be so kind as to point out where this happened. I have not seen it myself, but I am always happy to learn new facts.