RAMSTEIN AIR FORCE BASE, Germany — The 88-foot-long cargo hold of the massive airlifter has been converted into a makeshift intensive care unit. Metal stanchions hold collapsible stretchers fitted with heart monitors. Defibrillators, pumps, intubation kits, oxygen bottles and other equipment lie ready in their hardened cases.

For the scheduled seven-and-a-half hour flight between Afghanistan and this U.S. air base near the Pentagon’s Landstuhl hospital, the seven men and women of the Air Force’s 10th Expeditionary Aeromedical Evacuation Flight and their attached critical-care team will try to k...