Yes, they are agreeing to take charge and responsibility for the duration of the trip. Domestically from gate to gate, internationally from public areas to public areas. As RooFlyer pointed out, there's a separate request & agreement plus money paid for a commercially packaged special service. And at least most of us would have seen those UM's on planes (at least us using the cattle class at the back), seated at the very back under the watchful eyes of the crew.
I've sent my precious ones from Australia to Europe as UM's. The last time we saw them was in the international terminal at the entry to the security check & customs, where we handed the kids over to a flight attendant. Eventually they were handed back to the grandparents at the other end in the airport public area. While waiting for their connection in HKG, the kids were parked in a small airline office room with nothing much to do than read their books, exhaust the iPad batteries and talk amongst themselves.
The question is to be answered
how exactly could I be responsible, and specifically act, while the kids are speeding over Russia towards west while I'm on a different continent altogether?
For the kids in this Melbourne news story, while the "store room" was perhaps not the sweetest spot to camp overnight (apart from the forts built there
), at least it was in a controlled area and away from the general crowd at the airport or in the lounge. I'd consider that perhaps a "least bad" option given the situation. But the communications between the airline and family clearly didn't go to the plan but it sounds like it was a two-way street in this case. VA would have phone numbers for both the sending and receiving parties. The story does not tell how much they tried to contact the family but a few tries over time to both numbers should have resulted in success eventually and we wouldn't be reading of this at all.