Red Mites live in the roosting polls, in the cracks, and in the joints of your coop, then come out in huge numbers after dark to feed on your chickens. Treating the chicken or its plumage to control* Red Mites is a futile operation. Like the Terminator "They'll be back." You will achieve almost total control* however if you mix Diesel oil or used motor oil with Permethrin and use a paint brush to dab this mixture onto the roost polls, paint the wood parts, and fill up the cracks or cracks in the wood, you will not only kill the Red Mites that you currently have but you will prevent new Red Mites from recolonizing your coop. The choice is yours.
* In agriculture there are two ways to deal with pest.
Control means that you try to kill all the pests, managing pests means that you employ some alternative strategy like crop rotation to keep the numbers of pest down to some arbitrary level that doesn't represent a serious monetary lost.
Chickens have an oil gland at the root of their tail and they use the oil produced therein to dress and waterproof their feathers while grooming themselves. You can watch a chicken apply oil to its plumage by sticking its head into the base of its tail feathers then using its beak to spread the oil along its feathers. It looks somewhat like the bird is nibbling its plumage. This also is how a chicken interlocks its feathers to the adjoining feathers which not only provides a weather proof covering but help it fly to roost and hopefully fly out of danger.
Shampooing a chicken removes this oil and leaves the chicken open all kinds of skin problems.
Again, the choice is yours.
