In rough laymans terms, lets think of a single object in the game under a player's control.
What the server is receiving from an "uninhibited" high-speed connection: forward forward forward left left right up down left right right forward
What the server is sending to you over an uninhibited high-speed connection: forward forward forward left left right up down left right right forward
Same commands, but lets add some inhibiting (missed/dropped info due to malfunctioning equipment or a limited connection = lag).
What the server is receiving from a "laggy" connection: forward forward ____ ____ left right up down ____ right right forward
I believe, but can be wrong or behind the times on the way its being done in AH atm, but it has been known that in this game (and many other online "shooting" games) the server "smooths" out the flight of those on limited connection by essentially predicting/guessing the pieces of data that get inhibited from those slower connection for those of us with higher connection. So!... same commands still, but now we got some server guessing going on for the blanks so that their flying appears smooth and seemless to the rest of us (and not jerky/laggy every single time a bit of data gets lost, which is also beneficial because this data packet loss can occur for high-speed users too, so overall it "smooths" out the game):
What the server is sending to you over an uninhibited high-speed connection: forward forward forward forward left right up down down right right forward.
This is also why planes float off the ground and skyward for players on laggy connections while sitting on the runway next to those of us whos connection to the server are strong and solid. The server is predicting that the plane was moving in that direction and at that rate. Then when some data gets sent from that laggy player the server goes "oh, wait, no he's back over here" so skip/jump/lag-jerks back onto the runway where he's supposed to be.
As for registering bullet hits from you upon other players, that's all done on your machines end, with the hits being then reported to the server and then to the opponent's computer, which then on their end decides that the 5-cannon rounds you scored on the wing ripped the wing off, and then sends those damage results (the wing torn off) through the server and back to your machine, which then rips the wing off of your opponents plane on your end (thus the delay in shooting an opponents wing and it (or any other damaged part) flying off).
This is all mostly speculation from observation and is subject to whatever HTC confirms or denies as they see fit.
So, to answer your question, they don't have to be moving at all, they just need their connection to skip/lag a bit and the server then reacts by plugging in its own "predicted" bits where the blanks are.