Gates can be moving, but not at faster than light. SG1 had the bit where they tossed an active gate into a sun to blow it up, and Atlantis had straight up orbital space gates with stabilizer rockets on them

There was some explanation about how the gates had to be calibrated to a general part of space in order to participate in the gate network which is why they couldn’t use it in hyperspace. That, and probably some physics mumbo jumbo about why wormholes can’t connect from within subspace or whatever