When it comes to cooking styles, each island of the Caribbean has its own unique variation on the Caribbean norm.
An island's colonial heritage can certainly have an effect on its style of cuisine; in the French West Indies, for example, you're likely to encounter both classic and modern French cuisine as well as fine wines from Bordeaux and Burgundy. But you can also find regional favorites with a local twist: a rum specialty or a tropical fruit dessert, for instance, may be featured alongside traditional "old world" items. Chefs in the Caribbean will often infuse old world cuisine with local specialties and Creole flair.
Some words of caution: Be prepared to dine at a leisurely pace in the Caribbean. The easygoing lifestyle of the islands carries over to dining, which will give you time to savor the authentic cuisine. Many Caribbean foodstuffs are enhanced with local chilies and other exotic ingredients, which your palate may or may not be accustomed to. If you overindulge, the best way to "put out the fire" is simple bread and butter.