Once upon a time the parents/guardians would hand the UM over at check-in to a staff member designated for the task; that person often took several in charge at that point. It was his/her responsibility to stay with the UM(s) and hand them over to the cabin crew. Signatures were required at every stage, I think.
The UM wore a large colourful plastic envelope on a string round its neck, containing its travel documents. It was told that removing this would result in execution.
If you saw a stray UM you had to take it in charge and get it to the right place. (Nowadays, of course, that would result in arrest and vilification as a paedophile, so I guess that's no longer required).
Cabin crew had to make sure the UM(s) survived the flight with no visible bruises, and that they were fed and watered, didn't injure other passengers and stayed in the cabin. They then handed them over to a designated person on arrival, getting a signature for each one.
Not having had to deal with or indeed dispatch an UM for a very long time I don't know if this routine is still followed. One of the earlier "Airport" series had a very funny episode where a passenger agent spent the entire time wandering about the terminal bleating to anyone she met "I've lost me 'UM. Have you seen me 'UM?", so I guess it hasn't changed much apart from laxer discipline.