Snarky Puppy is free to roam, and that’s the way that bassist/composer Michael League likes it. The instrumental band he launched in 2004 with fellow jazz students at the University of North Texas continues to expand its crisp fusion palette with gospel, funk, R&B and world music.

Snarky Puppy’s February album/DVD release Family Dinner, Volume 2 manages to span live collaborations with folk-rock godfather David Crosby (“He’s a character, but we love him and he loves us,” League says), Afro-pop master Salif Keita, celestial British soul siren Laura Mvula and Swedish folk outfit Vasen.

“It’s all music,” says League, 32, touting his group’s experience as session players for artists from Justin Timberlake to Snoop Dogg to Chaka Khan. “Every guy in the band is like a sideman, so for us to go from wearing one hat to another quickly is kind of part of who we are anyway, not only as Snarky Puppy, but as individuals.”

Those individuals change from tour to tour and show to show, based on availability, though core players usually include percussionist Nate Werth, trumpeter Mike Maher, saxophonist Chris Bullock, keyboardist/trumpeter Justin Stanton and guitarist Bob Lanzetti.

“It started out as necessity because I didn’t want to say no to gigs,” League says. “I’d take a gig and call the guys, and someone would be unavailable and we’d call the next guy.” Now, about 1,200 shows later, “Guys who have played the least number of shows have still played hundreds,” he says, predicting that nine players from a rotating pool of about 25 musicians dubbed “The Fam” will hit House of Blues on May 4.

All this may not seem like an attractive formula for winning Grammy Awards. Yet Snarky Puppy just snagged its second Grammy, taking Best Contemporary Instrumental Album with Sylva, a project with Dutch big band the Metropole Orkest. Snarky Puppy won its first Grammy in 2014 for Best R&B Performance with “Something,” a Lalah Hathaway vocal feature from the first Family Dinner volume.

“We all did our thing without the help of the music industry at large,” League says of his Brooklyn-based group’s rise through word of mouth. “We were just a band in a van and a trailer, driving around and playing the Thirsty Hippo in Hattiesburg, Mississippi, and any anonymous club in any anonymous town. That’s how we did it. There was never any element of people in suits sitting around a table saying, ‘So exactly how do we market these guys?’ ”

They even shied away from capitalizing on their initial Grammy hit, avoiding reviving “Something” live. “Let’s create new moments, not recreate old ones,” League says. “I don’t think anyone in this band likes living in the past. Also, we don’t want to sacrifice artistic quality. I don’t want to play that song—something that she [Hathaway] completely destroyed— with another singer. Why would we?”

Plus, fans can watch Hathaway kill the original performance online, an outlet for promotion that Snarky Puppy utilized to build its audience. “People around the world really feel like they know you by seeing you on YouTube,” League says. “When they go to a concert, they’re already familiar with how you look when you play, and that’s actually a surprisingly powerful thing.”

League says the band is most in its element during live performances. So when they decamped to a Texas ranch studio to make their April 29 release, Culcha Vulcha, he says, “Doing a traditional studio recording was the unnatural thing.”

Nonetheless, the bandleader says that Snarky Puppy quickly settled into a more patient zone for Culcha Vulcha, which floats into lush, flute-wound melodies in “Semente” and slinky slide-guitar atmospheres in “Gemini.” The goal, he says, was to “build on soundscapes in a bigger way” than they’re able to when they play live.

Now they’re back to the road, on a tour that includes a city that’s familiar for Snarky Puppy, not only for the occasional music-college clinic, but for League’s visits with his brother and three cousins. “Boston definitely has a lot of family.”

Snarky Puppy plays House of Blues on May 4.