Hollywood being what it is, “This Is Us” star Sterling K. Brown has spent years deflecting roles to yet again play “a Black guy in a white family who’s finding his way in.”

Brown said no because he’s adamant about not getting pigeonholed, as he demonstrated this year by shifting to “Biosphere,” a post-apocalyptic two-hander with Mark Duplass, and then “American Fiction,” which is simultaneously a family drama, a meta-literary satire and a commentary on race in America today.

“I wouldn’t want to do three movies in a row with the same kind of role,” he says. “The industry wants to say what a Sterling K. Brown type is but when they think they’ve got it locked down I like to mess with that.”

In “American Fiction,” Jeffrey Wright plays Monk, a repressed writer and professor who must knock down his own walls to rejoin the living; Brown is his brother,...