Yes!
We did a B2B 4-night/3-night where just DH and I cruised first and then our DS, DD & DSIL joined us for the second cruise. DS was booked in our cabin (which we kept both sailings) and DD & DSIL had a separate cabin. We had already linked our dining reservations for the second cruise and it was no problem at all to keep the same servers.
DH and I debarked on our own, went through customs, checked back in, and then waited in the Concierge Lounge to reboard. Our kids arrived at the port just prior to opening and were almost the first to come up the escalators. Since we hadn't reboarded yet, we were able to join them as they checked in, and then they were allowed to board with us, before concierge. Another family had a somewhat similar situation: the couple was divorced and the mother cruised first with the children and then the father cruised second. The kids were pretty young, and the father was allowed to board with his children as B2B'ers when we did.
We embarked for an 11-night Mediterranean in Barcelona. After checking in and boarding the ship, everyone who wanted to was allowed to debark on their own to tour the city, just as you would at a port of call. Quite a few passengers took advantage of the extra time in Barcelona. We just all had to be back onboard in time for the Mandatory Assembly Drill. You would just go ahead and exit after going through customs, enjoy the city, and then check back in at the port for your second cruise. Just remember to take all important paperwork and passports with you.
Have a great cruise!