Once the last battle has been fought and Rand (most likely) wins, what do you suppose the asha'man will do with themselves?

my guess is some type of Military Police. hunt down DF channeler survivors.

I'd guess they branch out a bit, some remain as a military force, or each of them serves a certain time in the military, then they branch out to healing, research, etc. like the Aes Sedai (though hopefully not too much like the Aes Sedai).

they are eventually going to be reintegrated with WT and there was one of the foretelling or a dream to that effect. where the windowpane has both dragon's fang and white teardrop. like in ancuent symbol of AS

eventually join the aes sedai and complete the circle