Every ride handles the swap differently. Ask the CM at the entrance when you arrive. They will tell you how they handle it. Most give a special Fastpass that is good all day for up to 3 people. Some have you all go in line and swap at the ride and some have you wait at the exit and swap at the exit.
This is what we do as an example. We are a family of 4 with a little one who doesn't like thrill rides. We first go and get Fastpass for the ride; we get 4 and use the little one's fastpass for one of us, RnR for example. Then I go with my little one and ask about baby swap. They give us the special Fastpass pass (again, good for up to 3 people all day long). I go and ride with older DD with 2 Fastpasses. Then we swap and DH rides with older DD with the other 2 Fastpasses. Then we use the special swap Fastpass for older DD to ride yet again with one parent who wants to ride a second time. In the end, older DD gets to ride 3 times this way.
Now if we didn't get Fastpass for the ride, then one parent goes in Standby line with older DD and they ride, then we swap parents and with the special baby swap Fastpass, the older DD rides again with second parent with that special pass (it's good for up to 3 people).
Hope I didn't confuse you. If you get the special parent swap pass, it's good for 3 people max.