You are right it is not unusual for a bonded pair to try and re-establish hierarchy rules in a new territory and they can do this by mounting etc which can cause a few tuffs to fly too. It doesn’t really sound like fights as you describe it, but chasing with a few tuffs pulled out during?? You do have to watch that it doesn’t get out of hand. But in general if they get along, and your male is telling your female he is the boss my getting her to back away from the food but no real fight breaks out— just a chase in which the female doesn’t fight back, then it’s probably okay. Just keep an eye on it.
In the past, Jack always had to make it clear he was the King by making Vivian leave a litter box whenever he wanted in or he would make her leave a fresh pile of greens sometimes. If she didn’t leave, he’d try to mount her and then she’d leave. That was the end of it. If she decided to fight back then that could have been a problem.
It sounds like your female gets chased away, and maybe the males grabs a fur on the way? If that is it, then just watch it to make sure it doesn’t escalate as they work things out.