I never quite understood the option in the air field orders for carrier planes whether they should stay or move with the carriers.
Carrier planes operate in two ways.
First, they will automatically join naval battles in which their carrier takes part. You don't have to do anything to get them to attack.
Second, you can treat a carrier exactly as you do an airbase on land. Open up the icon, select air wings, assign them missions, and so on.
As with a land airbase, coverage of an air zone matters to the efficiency of your carrier planes used this way. This matters more in the Pacific and Asia than in does in Europe. The air zones are larger, and so longer range is more important. So, you're likely to care where your carriers are, and where they're allowed to move. If your goal is to use divebombers (CAS) to support island invasions or ground forces, or get some use out of the fighters for air superiority over land, you're likely to want to just park the carrier in a sea province near the land zone you're operating in. But you also might have your carriers patrolling the seas, looking for enemy convoys or fleets. You might want carrier planes doing air superiority over sea zones (as opposed to just cap over the carrier itself, part of that automatic combat). So, just like a land base, you might open up the carrier and assign fighters to that mission.
But then, you might have assigned the carrier to patrol multiple sea zones. The carrier will move away from the zone to which you assigned the fighters. So you have a choice: do the fighters keep patrolling where you told them to ("defend this spot here") even though the carrier is sailing further away (and thus lowering their efficiency)? Or did you really mean "fighters just patrol whatever sea zone the carrier happens to be in while it patrols around these three zones", even if that means leaving their original sea zone empty? You'll have better efficiency because the planes are closer to their base, but you won't be patrolling in that sea zone all of the time, either. If you want the first behavior, tell the planes to stay put. If you want the second, tell the planes to move with their carrier.