TMs: One innovation the White man inherited from the natives was the use of TM (T-something Medicine), a mix of herbs and spices, all thrown in with a bit of a hallucinogen that only the natives know how to grow, when consumed, the TMs force a pokemon to go on a dream journey, all alone for three hours. In those three hours they will not be able to move or fight on their own as they will be tripping balls. When they wake up and snap out of it, they will then know the move that the herb mixture promised.
HMs: Not so much a bit of medicine as it is a shamanistic practice. The holder of the HM has no physical item, he instead hold the knowledge of the HM's move, and can force any pokemon that is able to onto their dream journey without needing to dose them with herbs and peyote.
Alternatively, it IS a physical item which is imbued with the power of the TM which can be used an unlimited number of times IN a Shamanistic practice. Whichever one works better.
Pokemon that cannot learn the move but eat the TM anyway get sick. They will be useless for three hours, having bad dreams, and will lose a lot of HP. If left unattended they can die.