For the second year in a row, Jeffrey Tambor won an Emmy for playing Maura Pfefferman on “Transparent.” The Amazon series is about a father of three who, once his kids were grown, transitions to a woman.

Tambor is exceptional on the show, and he made a big statement when he accepted his award. First he gave an impassioned plea to Hollywood decision-makers to give transgender talent a chance. Then he said, “I would not be unhappy were I the last cisgender male to play a transgender character on television.”

His statement was just the latest in a recent push to change the way movies and television shows are cast.

From newcomers like Rami Malek to classic standards like Julia Louis-Dreyfus, here are the winners and other highlights from the 2016 Emmy Awards. (Nicki DeMarco/The Washington Post)

For many years, the surefire way to win an award — whether it was an Emmy or an Oscar — was for a well-known actor to play a character with a disability. It’s worked again and again, whether it was Daniel Day-Lewis in “My Left Foot,” Dustin Hoffman in “Rain Man” or Claire Danes in “Temple Grandin.” In each of these cases, able-bodied actors delivered the tour-de-force performances.

For the first time, there has been some very loud pushback against the trend, especially when it’s so difficult for actors with disabilities to get work. When the movie “Me Before You” came out this year, the quadriplegic protagonist was played by Sam Claflin. There was a major uproar from disability activists.

“Othello was [once] played by Laurence Olivier in blackface and we would never think in this day and age of casting a white person to play a black person,” Ruderman Family Foundation president Jay Ruderman said at the time. “Yet routinely people with disabilities are played by able-bodied people.”

Transgender characters have suddenly started appearing on-screen a lot more, but the casting of these characters follows the same trend as the characters with disabilities. In addition to Tambor, Jared Leto won an Oscar for playing an HIV-positive transgender woman in “Dallas Buyers Club.” Meanwhile, few people can name a transgender actor off the top of their heads, aside from perhaps Laverne Cox, who was nominated for an Emmy for “Orange is the New Black” but didn’t win.

“Transparent,” like “Orange Is the New Black,” has won plaudits for giving audiences some idea of what it’s like to be transgender. It also employs transgender writers, which doesn’t happen very often. There are transgender actors working on “Transparent,” including Trace Lysette and Ian Harvie. But, as Tambor noted in his Emmy speech, the meatiest role went to a man who has always identified that way. If even he thinks that’s a problem, the industry should probably pay attention.