I have had many children that MUST be handed from parent to myself or my mother. I get so frustrated when my mother is pre-occupied and I'm busy and think she'll pick up the child. Rarely now, she'll open the door and just stand back like that child is going to be put down. Eventually, they can be. But when is different for every child and bless God, I expect my mother to know which ones should be picked up!
We only care for 4 children at a time. BUT, we have different kids on different shifts and with the part-timers or afternoon betweens, we open the door between 7 and 10 times to let kids in and obviously, that many times to let them out. Imagine for a moment doing that many times over. The workers at your daycare should work better together. I've had to insist on my mothers being more actively involved with morning drop offs. Sometimes she prefers to take someone to school or change some dirty diapers than to greet some of the kids and talk with the parents.
I think you need to talk with the director about having just one main person, 2 at the MOST responsible for his transition times. If they are too big or too under staffed to do so, find an in-home provider instead.