Pace and others got it pretty right.
The AOC holder simply supplies the mode of transport and satisfied the consequent reasonable aviation safety protocols, it is the medical team who have the legal responsibilities for the patients health and wellbeing enroute. For routine transfers a responsible medical team will make a careful assesment prior insfoar as the patient is sufficiently stable before they accept medical responsibility and until this is transferred to the destination facility.
For real life threatening emergencies they must then make the difficult call the same way they do with regular ambulances in whether and/or when they move the patient at all.
It all gets messy if the aeromedical patient, (as distinct from a regular ambulance), expires en route which rarely happens as a proportion of flights, there is then the question of coronial jurisdiction. Not such a problem if it is intrastate but becomes so if interstate or overseas, given the differing legislation and protocols.
I suspect in these cases the patient is still technically "alive" when he is wheeled into the hospital emergency area. Keeps everything simple.
The elephant in the room is always the family who are desperate for their kin to be right, but have the potential to be pit vipers if it all goes wrong.
What does concern me and ther is a example extant at the moment is that the regulator appears to allow a lower standard of operation for medevac than for passenger carrying operations. Daily they, the public we consider to be uninformed, are being carried in non transport category equipment. my question I guess is are they or their family being informed that the equipment being used does not meet the same standards of the regular "public" transport. The regulator at last is moving towards removing the diferences and increasing the standards towards RPT, but in the meantime?