LOS ANGELES—Earlier this summer, the producers of a coming “Jumanji” remake put out a call to talent agencies: They wanted a Chinese actor in their movie.

Male or female? It didn’t matter. And what was the role, exactly? That wasn’t clear, either.

“They want to have a Chinese component. They don’t necessarily know what it is,” said one talent agent.

It was yet another example of a new Hollywood ritual—finding Chinese actors to cast in U.S. films to try to appeal to audiences in China, which is on track to become the world’s largest box office in the next couple of years.

The tactic has yielded mixed results.