The hottest Democratic presidential candidate right now is Pete Buttigieg, and the newest one, at least potentially, is Michael Bloomberg.

They don’t have much in common. They are separated in age by four decades, in hometowns by 700 miles, and in personal wealth by billions of dollars. But they do share this important trait: experience as American mayors. Mr. Buttigieg leads the midsize city of South Bend, Ind., and Mr. Bloomberg spent three terms as mayor of the megalopolis of New York City.

In...