Then, unfortunately, at best, they're getting Italian flavored feed, the amounts of helpful components in dried oregano are little but can vary. At worst, all that is happening is continual feeding low, low doses of antibiotics. Natural or not, it's still antibiotics.
If you ever need to use...
They will not help necrotic tissue, which is the most pressing concern. Oregano oil, the stuff that is actually good for infection, needs a carrier oil or it burns.