Editor's note: We thought Lovett Williams could answer this perfectly. Here's his response.
The culprits are likely moths of the type that also eat wool clothing. Roaches can cause damage, too.
Keep the mounts where they can be easily inspected, and watch for tiny holes in the feathers or evidence of feather particles falling off. When slight damage is detected, spray both sides with a roach-killing spray. Or better, spray them lightly once a year anyway. Be sure to spray the base of the tail near the dried flesh.