There is no automatic right to live and work in another Caribbean island because one holds a St. Kitts and Nevis passport. You will have to apply for a work permit to live on another Caribbean island. No health care is not provided in the other Caribbean islands.
With a passport of St. Kitts and Nevis, can I live and work in other Caribbean countries?
I am interested in investing in St. Kitts and Nevis to get a passport, but aside from the travel benefits, I am wondering what citizenship might get me. Does it provide health care on the island? What about benefits in other Caribbean nations?