No sooner had the curtains closed on the 88th Academy Awards, which was punctuated by Chris Rock’s scathing but broadly praised opening monologue on racism in Hollywood, than a new battle had reached a fever pitch on social media. And it had its own hashtag: #notyourmule.

Some viewers took issue with what they saw as the narrow focus of Mr. Rock’s opening monologue, which skewered racism in Hollywood but, they said, ignored the concerns of Asian, Latino and other minority artists. Others slammed the comedian’s bit involving Asian children posing as accountants as reinforcing negative stereotypes.

On Twitter, a brawl quickly erupted, highlighting a seldom-discussed type of racial tension: Does one minority group have to fight for another?