The U.S. military didn't always bring home its dead. In the Seminole Indian Wars in the early 1800s, most of the troops were buried near where they fell. The remains of some dead officers were collected and sent back to their families, but only if the men's relatives paid all of the costs. Families had to buy and ship a leaded coffin to a designated military quartermaster, and after the body had been disinterred, they had to cover the costs of bringing the coffin home.

Today, air crews have flown the remains of more than 5,000...