Lin-Manuel Miranda is certainly having a moment - a very long Moment. We have Hamilton on stage and the movie adaptation of In the Heights now in cinemas. To come are, among other things, an animated musical, Vivo, with songs and vocals by him and he's also writing additional lyrics for a live-action/CGI remake of Disney's The Little Mermaid. No doubt a full-scale movie version of Hamilton will appear eventually.
It's an impressive achievement, becoming a multimedia pop culture fixture in so short a time. Part of this, of course, can be attributed to technology, since there are more platforms with wider accessibility than ever.
Other songwriters for musicals had to wait to get anything like the same level of success. Rodgers and Hammerstein had stage and film hits from the 1940s (they won an Oscar for the song It Might As Well Be Spring from State Fair) and several of their musicals were made into films in the 1950s and '60s, with The Sound of Music being the biggest hit. It's unlikely their less popular shows - Allegro, Pipe Dream, Me and Juliet - will become films and some of the others might be too racially charged.
Lerner and Loewe's big moment came with the successive hits of the stage version of My Fair Lady - later sold for a handsome sum to Warner Bros who had a movie hit - and the Oscar-winning movie musical Gigi. The latter did not do nearly as well when revamped as a stage show. Camelot was a troubled stage musical and a disappointing movie but the association with the JFK presidency and some good songs redeemed it somewhat.
Stephen Sondheim has had a few of his shows filmed - as lyricist, Gypsy and West Side Story (the latter has a Steven Spielberg remake coming up: Gypsy could use a remake, too). As for his shows as composer and lyricist, we've seen some disappointments - A Funny Thing Happened on the Way to the Forum and A Little Night Music - and some better efforts (Into the Woods, Sweeney Todd). We might see other attempts.
Andrew Lloyd Webber's movie musicals have been uneven - Phantom of the Opera did not repeat its huge stage success and Cats was a massive critical and financial disaster. But Jesus Christ Superstar and Evita have their fans. A film of Joseph and the Amazing Technicolor Dreamcoat might be fun though Sunset Boulevard will always be in the shadow of its classic source film. Maybe he could try his hand at an original movie musical?