Well, I think the assumption is that special powers only have effect when they're being used, which is usually only during that figure's turn. Exceptions being special powers that explicitly state they trigger during another figure's turn. But Flying says nothing about being active during another figure's turn. In fact, it explicitly says it applies for "X's movement", which implies that it is during the "move phase" of that figure's turn. Personally, I believe that making the assumption that Flying is active during another figure's turn would set a precedent for other strange effects - for example, you could then argue that Elektra's Phantom Walk is active during another figure's turn, and that she could therefore be moved through other figures when being knocked back.
In terms of Classic Heroscape, I think the fact that certain powers explicitly point out the fact the flying figures will not take damage implies that there is no such general rule in effect, which is why those powers felt the need to call out the exceptions.
At least, this is how I've always interpreted it.