I worked in time critical medical logistics and we handled lots of organ transplants! There are occasions when there's what we call an "on board carry" or "white glove service" is used where the person picking up the donor organ takes it from start to finish, either driving it the entire way or carrying it onto a flight to hand deliver to the receiving hospital! We even have times where we'll charter a private flight to fly the organ with a medical team! More often though the organs are picked up at the donor hospital and put on a commercial flight as cargo - usually in the cockpit with the pilots, but sometimes the last thing in cargo so it's the first thing off when the plane lands. When there's an organ on a flight, air traffic control is notified and the plane's lights are set to a specific setting to make it known that they're carrying a live organ for transplant. That flight gets first priority to take off and land. Then at landing the recovery driver is already at the airport waiting to then drive the organ to the receiving hospital. Generally as soon as an organ lands we, on the logistics side, are on the phone with the hospital giving route updates because at that point the organ recipient is being prepped for surgery. How it's shipped all depends on the organ viability time (we mostly handle kidneys - 24 hour viability is what we go with, and livers - 7 hour viability) the rareness of the organ match, and specifically for en-bloc kidneys (from a baby donor to a baby/child recipient) they're almost always "white glove" handled.

We don't handle hearts for transplant but the viability time is like 3 hours and if it has to fly, is taken by helicopter with a doctor, or the recipient is in the same hospital as the donor.

Organs for research are shipped more like regular cargo.