“The Late Show With Stephen Colbert,” which premiered Tuesday, has local connections. The house band, Stay Human, features two alumni from the Boston-based program “From the Top,” which highlights the work of young, classically trained musicians. (NPR’s show “From the Top With Host Christopher O’Riley” launched in 2000.) The former “From The Top” stars who are now playing for Colbert are tubist Ibanda Ruhumbika and bassist Michael Thurber. Thurber is now a board member of the organization, and thanks to Ruhumbika, Stay Human, with bandleader Jon Batiste, performed at “From the Top’s” 2013 gala. “If you want to be a classically trained musician in the 21st century, you kind of need to know how the world of professional music works. . . . There are all these classically trained musicians out there, who are finding new ways to make careers,” said “From the Top” radio and concert producer David Balsom , referencing the group as a successful example. “I have a feeling that . . . music is going to be a core element as [the show] goes forward.” Other former “From the Top” musicians include concert pianist George Li, tubist Carol Jantsch, and cellist Oliver Aldort.