As with an extract or bomb, a mutagen that is not in an alchemist’s possession becomes inert until an alchemist picks it up again.
Can a non alchemist take a mutagen and carefully pour it into the alchmist's mouth as is possible with a potion?
If it is a mutagen that increases con it could help stabilize the alchemist. Or if he is a bramble brewer in bright light he would gain fast healing.
As long as the non alchemist is holding the flask it is inert, but does having it in your mouth count as having it picked up?
By the strictest interpretation of the rules, an alchemist's abilities are caused by both the substance/bomb AND magic in their own body.
Without bother, it doesn't work..
One would assume an unconscious alchemist isn't exuding his magical forces that make all his stuff work.
As a DM I personally threw that idea out the window though and made alchemist's stuff work like actual potions.
But I am a lot less concerned with class balance as much as I am concerned with the enjoyment of my players.