The very first thing i thought of when i read the first post was that my dogs only tend to dig when they smell/hear something. Perhaps you have termites like another noted, or even mice or smaller rodents could be your culpret. If the damage is done i would go ahead and cut a 1 x 1 square where they were digging and take a look inside the wall to confirm no rat poop or something along those lines is inside the wall. You could even check to see if the wood structure behind the drywall has termites while your looking. Then just patch the wall right back up. Damage is already done. It could only lead to a possible solution.
Good luck though. Atleast drywall is cheap and easy to fix!