“There is a different sensibility with regard to these issues, so we wanted to respond,” said Ellen Sorrin, the Trust’s director. “We are not demanding that companies do this; we’re making them aware that these are concerns.”

These adjustments are part of a broader effort to re-examine how people of color are portrayed in the performing arts and how classics with potentially troubling aspects can be made acceptable to modern audiences. In 2015, the Metropolitan Opera eliminated blackface from its “Otello.” The Bolshoi has toned down a segment of its “La Bayadère” featuring white children in blackface, but it has been criticized for not going far enough. And more recent fare has also been revised: The musical “Cats” dropped a song in which characters sang in Asian accents.

“I see what’s happening at the Balanchine Trust as part of a larger movement in the arts to address entrenched cultural narratives that undermine tolerance, fairness and inclusion,” said Lane Harwell, a program officer at the Ford Foundation, who previously served as the founding executive director of Dance/NYC, a service organization. “Any resistance is just because change is hard, but performing arts are not a static form, and continual interrogation and reinterpretation keeps historical work relevant.”

There has been some pushback from purists who argue against altering a work of art in response to changing times — and there is particular sensitivity surrounding Balanchine, whose “Nutcracker” is a widely cherished tradition. But, for the most part, ballet companies around the world seem to be on board.