Here's how normal child swap operates:
It depends on the ride. For the sake of this post, I'm assuming we're talking about a height restricted ride like splash mountain. Disney finds it strange if you try to child swap on rides like buzz lightyear or toy story mania or haunted mansion. Also, I'm assuming that the ride is a fastpass capable ride.
So, when it's a height restricted and fast pass ride, the whole entire intact party goes up to the cast member at the entrance (who is preventing entrance to those who are too short). The cast member says: "Who will be riding and who is waiting with the child?" The CM hands the waiting adult a fast pass equivalent called a child swap ticket. The party separates and meets up after the riders exit. The person waiting with the child swap can then go with up to 2 other people into the fast pass line and present the child swap pass.
There's no special entrances for child swappers, and there no secret entrance. There's no using the handicapped entrances or entering through the exit.
I'm not sure how they would do it at the haunted mansion (which doesn't have fast pass) or at Pirates of the Caribbean or other non fast pass rides. (I don't remember any height restricted rides that are not also fast pass.)
That's the basic use of the child swap ticket.
For the more "advanced",
I like to call it the "child swap cheat". If you "collect" child swap fast passes, you can use them later in the day. You don't have to immediately swap.
Here are the ins and outs:
A child swap ticket operates just like a fast pass, but you're not restricted by the fast pass machines on getting them.
To obtain a child swap ticket, you need at least 2 adults and one child. All 3 must go up to the cast member distributing child swap tickets. The cast member will ask: "Who will be riding? And who is staying with the child?" At this point you SAY one of you is riding. The "rider" enters the standby line. And the other adult obtains the child swap. Voila, you now have a child swap ticket. (But nobody has to stay in the standby line for hours). You can do this at various attractions starting around 10:00, and you'll have a ton of these "fastpass equivalent" passes known as child swaps to use during peak hours of noon to 4.
The Disney intention is that you use it right away.
But you can hang on to it to use that day. I think some of the child swap tickets are not even dated. You could use those later in the vacation.
After obtaining 1 child swap, go to a ride that you all like (say country bears), then with child swap fast pass in hand, you can go back to splash mountain and do the same exercise above and get another child swap. Then you'll have 2 child swaps.
Using the technique of obtaining a child swap ticket twice, no one ever really has to wait in the standby line of a height restricted ride when you have a toddler.
It actually works for rides like toy story mania too. But you can't use it right away. It's really only better than a fast pass in this instance because fast passes typically run out for that ride.
But they're really watching you hard for funny business at toy story mania, splash mountain, soarin, and some other busy places like expedition everest. But it's really easy at rocknroller coaster and tower of terror, space mountain.
In a pinch, if I'm hassled I just get in the standby line with my child swap pass, hop over the ropes to the fast pass line after the first "greeter" and hand in my pass at the front of the line. This doesn't work where the two line are separated.
It's really nice to use the child swap passes if you have a third adult babysitter (like grandma) who doesn't want to ride space mountain. You just obtain the child swap (via process described above), exit the line, do something else or wait 10 minutes, and then up to 3 adults can go right over to the fastpass side.
I know it's borderline cheating, but I liked the old system better where the whole party waited in line together, kids and all. That was actually quicker than a second person waiting to go through the long fastpass line.
And it still takes longer for 2 cycles of waiting through fastpass than 1 wait in standby.