When I got my 10-year visa for China in my US passport, there was a note stapled in saying that it would remain valid after the expiration of the passport. So I'm not worried about that one.
Does anyone know how it works for India? Their consular website lists a fee for transferring an existing visa into a new passport, so it sounds like they make things more difficult… If that's the case I guess I'll continue using e-visas until I renew my passport.