The NPR podcast A Surgeon Reflects On Death, Life And The 'Incredible Gift' Of Organ Transplant features Dr. Joshua Mezrich a surgeon who flies to other hospitals to retrieve organs for transplant discussing his book When Death Becomes Life.

In the beginning of the interview he says (my transcription effort):

There’s lots of different ways different centers do it. At our center we have some pilots who fly for the state who are on call for us, and typically we fly in a Pilatus, which is a single engine turboprop,

It is true that organs sometimes get sent on commercial planes; kidneys in particular, so if you ever see a cooler in between the pilots when you board it’s typically a kidney.

Are there little seat belts or other restraints for the organ-containing coolers kept in cockpits of commercial aircraft? Or are they just placed on the floor?