They'll check you in for the next cruise while you are on the first one. On debarkation day for the first cruise they'll have B2B people meet someplace on the ship (they'll tell you), if they haven't already issued your new cruise card they'll do it at this meeting (the card for the first cruise is not used for both legs), and then if the local Customs officials require it someone from the ship will escort you off and you do a walk-thru with Customs and then reboard the ship. You will bring your passport, a Customs declaration for the first cruise, and the ship gives you a 'in transit' card which you show Customs. That's all. You really just do a walk through where they hardly look close and wave you by. Detailed instructions are provided before the first cruise ends. I don't know about Carnival but on RCI we had a meeting on the last afternoon of the first cruise where they handed out our new cards (so yes, I had two cruise cards for a day but only one worked) and provided a briefing.
You don't have to pack up and leave and then check-in again like a brand new guest. If you are in the same cabin for both cruises you just leave everything where it is. If you are moving to a different cabin the cabin staff will coordinate the move and they will carry your clothes and stuff to the new cabin for you.
It is a painless process.