It varies somewhat by country.
We haven't touched EVERY country in the
Caribbean, but we haven't yet seen one where arrival at Oh-dark-thirty required an immediate checkin...
...with the exception of
Barbados. Arriving boats there need to
contract port control on the
radio, and are directed to proceed directly to the
commercial dock, NOT go to the anchorage. It's a real PITA. But most cruisiers don't go there without a good reason, for good reason!
For every place else, just have everyone stay on the
boat until the
captain completes formalities to get cleared in. Some places are more uptight about this
rule than others, but you really don't want to push it. Do
not go touring until you are official.