I played Desperate Alliance to pass a hero to a friend, in quest phase, in 2b of Foundations of Stone. We past to 3a, in 3b each player starts an only way... question: the hero past by Desperate Alliance keeps in the new player's control? or return to my control?

That is so strange that you might ask Caleb directly (Rules Question at the bottom of the page).

I would say the Hero should stay with the player... but don't know if it must return afterwards.