I have seen a thesis on the migration of Irish to the Caribbean and Jamaica. The thesis makes the same observation that there are many people in Jamaica who claim descent from Cromwellian Irish emigrants. However census records of Jamaica do not show it as having a large poulation until later and few Irish in the time of Cromwell. The thesis goes on to conclude that in the first generation most Irish transports were actually going to Barbados. Many died and or left no children. The island of Barbados could not carry as many people with opportunity as lived there by the second or third generation. Most of the Irish descendants born on Barbados then migrated to the other British colonies including Jamaica.