Something looks strange here. Judging from the length of the nails alone it doesn't look like this dog is receiving the best care. How old are these dogs?
Boy that certainly looks like "Mange," to me?
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But fours Vet's? Someone should have found the answer? Have they at least had Kenalog shots? That's not the complete answer but it should provide some relief for them.
Most likely he is getting tired of trying vet's? But I would contact a Rescue Group, and ask them where they take there dog's with "skin issues?" Most likely "whatever" this is they have most likely seen it? Maybe he can arrange for someone from a "rescue group," to see the Dog's?
I would also start using "Food Grade Diatomaceous Earth," on the Dog's Bedding and in room they are in the house. Now I don't know if they have the mites? But if they do ... this will kill them and it fall's in the heading of ... "it can't hurt."
And also it might be "Food Related?" And if so just stay away from "Grain Free Dog Food!" He'd just e traiting one isues for another. And many on here know what I am alluding to.
Thank you for trying to help him and Thank him from us for "Rescuing them." Keep us updated on how it's going?