The rise to international stardom of Staff Benda Bilili reads like a very contemporary version of an ancient African miracle tale.

Just a few years ago, the band’s four original musicians—all childhood polio victims, and paraplegic—lived rough in central Kinshasa, often sleeping on flattened cardboard boxes. The teeming capital of the huge and turbulent Democratic Republic of Congo is a dangerous place for anyone with a disability to spend the night in the open, but there was little choice for them and other handicapés—as they’re known—unable to find work. Though the four guitarists were fine musicians and singers, none could get hired to play. So once a week, they’d pedal their hand-built tricycles to the grounds of the city zoo to make music that mixed Congolese rumba and soukous with reggae and old-style rhythm and blues.

Staff Benda Bilili—which means “look beyond appearances” in Lingala—was discovered by French cineastes Renaud Barret and Florent de la Tullaye in 2005.

“We rehearsed in the place in Kinshasa where the two whites [Barret and de la Tullaye] went to eat, and they heard us and said, ‘Oh, you guys work really well together,’ ” says drummer Cubain Kabeya, reached by cellphone on a bus taking the band to Boston on its first tour of North America. “In 2009, they brought us to the Eurockéennes Festival in Belfort [France] and everything happened from there.”

The filmmakers caught the extraordinary culture shock and collective ecstasy, both on-stage and off-, in their acclaimed doc Benda Bilili!, which premiered at the 2010 Cannes Film Festival.

The movie also shows how the handicapés are known for their education, loudness, fearlessness, and powerful “syndicate”—they form the second largest group of street people in Kinshasa—outnumbered only by the more than 40,000 sheges, runaway or abandoned children who somehow eke out an existence. Many sheges benefit from the protection and help of handicapés, like musician Roger Landu, who in his early teens created his one-string lute, the satonge, from a guitar string, a tin can, and a piece of wood.

“I came to know the group in 2003, and grew up with them,” the 24-year-old Landu says, given the cellphone. “They’d all heard about Roger, a street kid who lived in the open and played music. I appreciated what they did. I hung out with the group and really got to know their music. At one time I had the opportunity to get close to Ricky [Likabu, leader and singer] to learn from him how I could evolve, because it’s our old people who can do this. He said, ‘Come on, Roger, you need to find the right setting for your instrument, the right direction in which to take it.’ I didn’t expect to hear all the things he said. It was surprising to me. He said that I’d make it big eventually—tomorrow, or the day after. I was happy about that.”

Staff Benda Bilili’s debut album, Très Très Fort, was three years in the making. Its recent follow-up, Bouger le Monde, lives up to the punning French title—both a call for people to dance and shake, and a cry for movement and change in the world.

“In the past year, we’ve been everywhere—Australia, New Zealand, Nouméa [capital of New Caledonia],” Kabeya says. “We’ve played in Morocco and Japan. Now I can say we’ve really been around the world. We hope to go to South America next year to make the people there move and shake. It’s a dream—as the film shows, we’re real optimists.”

Staff Benda Bilili plays Club 560 on Sunday (October 28).