Right. As soon as a neutral is attacked, it immediately joins the other side, which means that during the noncombat movement following the combat move that was the attack on said neutral, aircraft can freely fly over the attacked neutral (bad way to say it - actually, it is no longer neutral at all, because it was attacked.). And then all powers from both sides can fly over that attacked neutral for the rest of the game. It has joined one of the sides and is no longer neutral.
Unfortunately, Triple A does not track this. You can do it with game notes or something.
You can NOT freely fly OVER the neutral during the combat movement phase that you are FIRST attacking that neutral for the very first time, as Bold said.