A new wave of research is showing what the old gym myths about stretching got wrong—and what you really need to do to reap its benefits.

Stretching is an essential part of almost every workout — especially running. Even going on a short jog works your muscles, and many doctors recommend stretching both before and after exercise.

A herd of toddlers runs across the springy gymnastics mat, each to a small square of carpet. Their parents follow close behind. Once seated, coach Marci Schuster takes them through some warm-up ...