We've done B2B on Divina, Seaside an Meraviglia. Our process has varied a little depending on port, whether in Yacht Club and if you are changing cabins or not. If it's a US turnover port you will have to get off the ship and go through immigration/passport control at some point. If not changing cabins there is no need to bring anything except passport, unless you are on a repositioning where the ship may take some time to clear and you could spend hours in the terminal waiting to get back on the ship.
When changing cabins the butler or stewards will move your packed suitcases, we take any valuables with us. On Meraviglia we met at the Carousel Lounge until all other passengers were off. Yacht Club passengers usually wait in the Top Sail Lounge until called to join the regular group. You use your old card to get off the ship and new card to get back on (not sure if card changes if you booked as one cruise). The process will depend on the port and mood of immigration officers. It has ranged from the line of us walking past an officer in the terminal and showing our passport to just being counted as a group as we stood beside the ship. B2B are the first to be let back on the ship after it's cleared.