An obscure Pentagon office designed to curb the flow of illegal drugs has quietly evolved into a one-stop shop for private security contractors around the world, soliciting deals worth over $3 billion.

The sprawling contract, ostensibly designed to stop drug-funded terrorism, seeks security firms for missions like “train[ing] Azerbaijan Naval Commandos.” Other tasks include providing Black Hawk and Kiowa helicopter training “for crew members of the Mexican Secretariat of Public Security.” Still others involve building “anti-terrorism/force protection enhancements” for the Pakistani border for...