See if you can get HEL or ARN both airports are good. And then you could use say Airbaltic to get them to BUD. Riga airport is excellent for meet and assist and has a very high English standard and Airbaltic look after their punters.
With this sort of thing its usually easier to reverse the trip so go to the BUD airport website and see which flights go out and then make plans backwards.
AMS although big is pretty solid.
I don't have a clue what the official method of dealing with PRM's if caught in a overnight delay once they are out of my care sorry.