When your transfer is approved, you will transfer with your boat and crew.
In some of the add ons available you have to actually make the trip yourself. In SH3 stock version, the game (with no realism at all) magically takes care of it for you.
What I like to do is to use some of my renown to upgrade to a new boat and do the same for some of my opfficers and men. This for me simulates a kind of promotion or perk in being given a transfer and a new command. That portion of the crew that I replace I imagine as being those members who were not allowed (or declined) to make the move with their heroic commander... me.
A big part is imagination and what you make of it. It you prefer to make the trip yourself then try one of the add ons available to force a more realistic transfer.