Before they take the field on any given day, the Mets might look less like a major league ballclub and more like a bunch of regular guys getting ready for a twilight softball game. This is in part because they have players both old (Bartolo Colon) and young (Michael Conforto), who come in many shapes and sizes. But it’s also because many Mets actually are softball players before games—at least, they are while they’re tossing a softball around instead of a baseball during warm-ups.

Outfielders Yoenis Cespedes and Juan Lagares,...