DE has a place and according to all available scientific research that place is as a filtering medium for Petroleum or crude oil.
All the initials FG or the words Food Grade means when discussing Diatomaceous Earth is that these tiny pieces of silica have been washed, rewashed, and metered...
That is a good lesson for everyone with chickens. A chickens' needs and a humans needs or druthers are seldom alike, what's good for one, is often deadly for the other.
The lye in ashes is the same thing that old fashioned homemade soap is made from. I would think that the lye in ashes would lite up lice and mites, but especially chicken lice because of their soft body.
Soft wood burns up almost completely so there is not a lot of ash produced. Hydrated lime, is high in base chemicals as opposed to acid chemicals. lye is also made by burning plain old lime otherwise known as calcium carbonate or hydroxide, traditionally called slaked lime, hydrated lime, or...
Hardwood ashes are high in lye. Lye is the basic ingredient of homemade soap. Lye is caustic and it is an especially good amendment to a dust bath to both prevent as well as to cure lice and mite infestations. I don't think that hard wood ash can replace either Permethrin or Pyrethrin but...
There is no evidence that Diatomaceous Earth a.k.a. DE is lethal to chicken mites and lice. Just like there is no evidence that Easter rabbits delivers colored hard boiled eggs on the first Sunday, after the first full moon, after the Spring Equinox.