Mark,
Very good question, but it varies significantly from one island to the next.
The French "possessions" are, technically, iirc, departements of France, and I believe that flights to Guadeloupe and Martinique are classed as French domestic.
The island of St Martin is split into the French side (Esperance has a very short runway and some local flights) and the Dutch, on which the infamous SXM airport is situated.
Of the world's 7 "divided" islands, St Martin must have the most relaxed borders, but customs and immigration are still handled through the Dutch Antilles rather than France, though AF have a higher frequency of service to the island.
The Dutch Antilles also include Eustatius and Saba - good luck getting a heavy onto either of those, that would be fun! Out of the ABC islands (Aruba, Bonaire and Curacao), Aruba is "semi independent", but all three are c. 1 hour flying time south of SXM.
UK crown dependencies in the region include Anguilla and Monstserrat. MNI has historical links with Ireland, but tourism is heavily restricted due to recent volcanic activity and the fact that the recently opened airport only has a 600m runway. Vast majority of flights via ANU (transfers previously by boat or heli).
Bermuda is also a UK crown dependency, as I believe are the Cayman Islands, but I'm not sure if either would be classed as "part of the EU" like the French islands, considering that none of the Channel Islands or the IOM "enjoy" such status.
I would expect that there would be more of a market for the "Anglophone" islands, where English is the first language, and would guess that there would be huge amounts of bureacracy to open up flights to any of them, but I'm sure there are a few locals who could comment in more detail.