When Melinda Roberts found shingles in her front yard after a storm, her insurer didn’t dispatch a claims adjuster to investigate. It sent a drone.

The unmanned aircraft hovered above Ms. Roberts’ three-bedroom Birmingham, Ala., home and snapped photos of her roof. About a week later a check from Liberty Mutual Insurance arrived to cover repairs.

“It took a lot less time than I was expecting,” Ms. Roberts said.

Drones, photo-taking apps and artificial intelligence are accelerating what has long been a clunky, time-consuming experience: the auto or home-insurance claim.

Traditionally, an insurance claim associated with minor home damage or fender-bender auto accidents started with a phone call from a customer and ended days or weeks later with a mailed check. In between the insurer often would send an inspector to investigate the situation in person.