Disneyland (my experience from Avengers and Star Wars): As PP said, runDisney organizes the kids by age in a section of the parking lot. (No strollers allowed in this designated section.) Then, runDisney walks the kids (by age group) to their start line. Then -- this is the part I haven't seen firsthand -- runDisney must release them on a staggered basis in groups of 10-20. So, once a particular age group starts, you have this continual stream of kids crossing the finish line until the end of the age group. Then they line up the next age group, and continue the process all over. The finish line feeds directly into the pick-up area, where the kids get their medal, water, and a snack. To get out of the pick-up area, runDisney was extremely vigilant in matching up the kid's bib number to the parent holding the matching, tear-away portion of the kid's bib number.
Disney World (my experience from Marathon Weekend): The non-mile races are along a track at WWoS. The kids line up near their respective start lines on the interior (grass) of the track. runDisney releases the kids somewhat orderly. Again, the finish line feeds into the pick-up area. In my experience, the pick-up area was NOT enclosed. So, while I meandered over to the finish, DS left the area looking for me. I wasn't worried, but I was annoyed walking in circles looking for him. At no time did anyone ever attempt to confirm whether he was my son. This seemed chaotic to me. DS also did the mile. runDisney tried to have 2 informal corrals based on where the volunteers were standing, but without a rope, it was difficult to enforce. The mile finishes in the same location as the other kids race, but I ran the mile with DS, so I didn't have to locate him at the end. Again, no one asked me to verify that he was my son.
After you finish, you get your medal, water, and snack, and then you're done.