WASHINGTON—Music thumped through Freedom Plaza one recent Monday as more than 100 spandex-clad figures held warrior poses under the unforgiving noontime sun.

"How do you tax energy, energy that's moving through—left foot forward, step" the event's MC exhorted, mixing political commentary and yoga directions. "Not taxing human beings for getting healthy—hands down low, push-up, up-dog, breathe in!"

Yoga...