In an 1876 book, John Solis Cohen said that Galen explained how the ancient physician Aesculapius blew "an astringent powder into the larynx by means of a bent reed" in patients who presented with angina, or chest pain. (1, page 336)
The medicines used were nutgall and myrrh.
So chances are pretty good other physicians in the ancient world relied on patients inhaling powders as remedies for various respiratory ailments (and probably other ailments too) all the way back to the beginning of civilization, and maybe even into the primitive world.
- Cohen, John Solis, "Inhalation in the treatment of disease," 1876