rofl!!!! Now that's the kind of humor I get!
You're going about it backwards. There's NO way to deter external parasites, none whatsoever. They come from birds flying over your yard, they're literally everywhere. What you do is make sure your birds are Ivomec'ed or Frontlined, that's all you do and it works.
1st you need to figure out if it's actually mites or lice. I would bet money it's lice. Lice are tiny sand/tan colored specks that crawl fast. Those are the ones you're going to see 1st, mites are even smaller and pretty difficult to see. Lice hang out all over the skin, the easiest place to see them are around the vent. They spend their entire life cycle ON the bird. Wooden or wire cages won't matter. LIterally ALL you need is Frontline or Ivomec. I know Frontline & Ivomec Eprinex (different than just Ivomec) works excellent, those are the 2 I've specifically tested on birds I got elsewhere. I guarrantee you-if birds aren't treated......they have lice (most likely) or mites (less likely).
There's generic Frontline now, called "Fiproguard" on Ebay. I got 3 huge vials labeled for 89-132 lb dogs for $20. Birds only need 3-10 drops (10 drops would be for a 10 lb bird), it comes to about a drop per pound. Keeps lice and mites off them for 3 months. You should read about it, although most people have their info wrong so decipher what you read lol I know I harp so much on worms and bugs on birds that no one really listens to me, but it's my particular interest. Some people like the egg aspect of raising chickens, some like the housing part, some like the butchering part. The part that fascinates me about birds (and fish and dogs and cats, etc etc) is the health aspect part of it.