I'm going on Liberty to Labadee, Ochos Rios, Grand Cayman and Cozumel in March.
The ships dock in Labadee, Ocho Rios, and Cozumel. The only tender port is Grand Cayman. When the first tenders leave depends on when you arrive and when the ship is cleared by local authorities. Bear in mind that you will need to get tender tickets (free) for the first several tenders. When these are handed out will be announced in your daily Cruise Compass. Space can be limited on the first tenders because they offload passengers who bought tours from the cruise line during this time and that can eat up lots of space.