Easy process as others has stated. Just make sure that your reservation indicates a b2b, the ship should catch it but best that they know upfront. They will leave you instructions in your cabin on what to do.
BTW-there are some circumstances that you do not have to go through customs on turn around day even in the U.S. If doing one from Miami I try to book the first cruise with a US port prior to 1st debark (Key West or St. Thomas) then you don't have to do anything.