If you track the margin of support for the winning presidential candidate in Florida versus the rest of the country, Florida only started voting more heavily Republican in 1960 — the same year the number of New York-born Floridians first surpassed 5 percent. There are a few possible reasons for this. The first is that the people who move to Florida tend to be retirees. While it hasn't always been the case, older voters tend to vote more Republican. That's particularly true right now, meaning it should hardly be a disincentive for the state Republican party.
The second is that New York was becoming more Democratic as Florida was becoming more Republican. If we're assuming that migration patterns affect political voting, it seems fair to assume, then, that it was more likely to be Republicans moving South — a bit of political self-sorting.
But the third and most important reason for Florida's voting shift is that Florida is a Southern state. Over the latter half of the 20th century, the Deep South went from dark blue to dark red. Georgia, Louisiana, Alabama, South Carolina — all states that moved away from the Democratic party even more than did Florida. The migration of people from New York was almost certainly a drop in that demographic bucket.