The process is you tell the person at the queue you are rider swapping, they will give you a lanyard to take into the Fasspass line. (those not riding wait outside). You give this lanyard to the next cast member you see (usually right before entering the boarding area- they then give you a printed pass ticket.
The ticket itself says up to three admitted for the ride, pretty sure up to the end of the day. The next party then gives that ticket to the cast member at the entrance upon which they can enter (no need to scan any more magic bands) So if you have two kids big enough for the ride, they can ride again, assuming they went with "Dad" first, and now they want to accompany "Mom".
This past trip, it was just my son (9) and daughter (3) with, along with my wife, so my son got to ride everything twice which my daughter was too small for (which he of course loved.), but in theory we probably could have brought anyone else along and they would get in, because of the "up to 3 guests admitted" indicated on the ticket.