I don't know about land based. Boat based the captain goes in with the passports plus departure cards for each passport plus the green sheet from customs.

When we cleared in a couple of weeks ago they screwed up. Customs thought we were checking in/out at the same time but immigration wanted us to clear out. Customs kept our green sheet when we checked in. So when we went to check out they wanted the green sheet again. I told them they already had it. They made me fill in the long form all over again.