Actually, DNA work was done (15-20?) years ago by Glen Collier and others, using reptiles, amphibians, and cyprinodont fishes, to show that the most of the fauna on Cuba and Hispaniola are relict lines that were stranded when the old land bridge broke up. I sent a copy of the paper to Willem a few years ago after seeing his talk at OCA, PM your email to me and I will send it to you. The larger islands were a part of of a land bridge, yes, but I'm pretty sure they moved east, and a new land mass moved in from the west to form the current CA. This one was colonized again from the south (so far as killifish and reptiles are concerned), I forget the exact timeline, but it was millions of years in the process. There was never any contact between the current isthmus and the Greater Antilles except where it came into contact with the two continental land masses millions of years later.
That paper also determined that Trinidad and most of the lesser Antilles were
colonized by SA lines during the last Ice Age, when ocean levels were very low. At that time they made up a peninsula, which was easily invaded from the mainland. There are several species on Trinidad that also exist on the mainland of SA, so the connection was fairly recent by geological time.
As usual, the cichlid researchers are a decade or more behind the killifish science. (Sorry, couldn't help myself.)
There are two kinds of error: blind credulity and piecemeal criticism. Sound skepticism is the necessary condition for good discernment; but piecemeal criticism is an error. - Egyptian proverb