OVER THE ATLANTIC OCEAN—The carry-on bags in the luggage bin above seat 7G on a recent U.S. flight to Dubai contained thousands of bills—$20s, $50s and $100s, all neatly wrapped in rubber bands and plastic.

The cash was carried aboard by Abdi Warsame, a Somali immigrant and an employee of a Midwestern money-transfer company that was shut out of the international banking system last year. The firm now ferries cash by hand.

The...