Remember that -hez is a movement that ends up -nél (at, by, next to), not -ben (inside).
So they move not all the way to the airport as in ending up inside it, but stop just before reaching it. Their house might be opposite the street from the airport, perhaps, which is still "by" the airport, and is possible with city airports such as the one in London or even Hamburg.
Yes, that is the idea. Or near the airport.
The English "to" does not exactly fit any of the Hungarian suffixes (but partly covers several of them), so these ambiguous/questionable translations are bound to keep popping up.
In this case, if the new residence were actually on the site of the airport, the question would be "A repülőtérre költözöl?".
My point was simply that "moving to a place" and "moving near to a place" are not identical constructs. When learning a language, less ambiguity, and greater clarity seems preferable. But that's why we provide feedback, and how we will eventually move this course out of Beta. Cheers!