The last time someone posted about a UM, it was because DL sent the two kids to the wrong city (they switched their paperwork and thus switched their destination cities). So it sounds like all's well that ends well and your case was a piece of cake in comparison

My guess (and it is just that) is that DL has responsibility for the child while in their custody (I am sure something is mentioned in the paperwork the person dropping him off at the airport had to sign). They would probably put him in a hotel but I would think you would be responsible for the hotel cost (unless it was mx related).

One reason that the airlines don't allow UM's to travel on the last flight of the day is just for this reason - it gives the airline at least one more shot at getting him/her to the destination the same day.