Few viewers came out of Pitch Perfect 2 thinking, “this franchise will run and run”. Second time around, the premise of competitive a capella already seemed to have exhausted its possibilities. How many more “riff-offs” did we need to see? How many more big competitions were left to enter? How many more slightly over-extended Rebel Wilson one-liners could we take? But this third – and surely final – outing basically explodes its own formula. It’s like a good Christmas pantomime. It assumes we all know the drill, then has a whale of a time subverting it. In the process it throws out all semblance of plausibility, but by this stage, who really cares?

The opening set-up is literally explosive: the Bellas are on a luxury yacht, performing another of their choreographed, beatbox-backed cover versions for the delectation of three unknown men. Suddenly Fat Amy (Wilson) comes crashing through the skylight, hoses the men with a fire extinguisher, and they all jump overboard before the yacht explodes.

The “How did it come to this?” backtracking that makes up the rest of the film packs in a great deal of incident and a great many characters. The older generation of Bellas, led by Anna Kendrick’s Beca, are struggling to cope with the outside world, post-university, and a show by their perkier successors only makes things worse. So of course they leap at the totally spurious opportunity to reunite and embark on a tour of European US military bases. “Is there a competition? There’s always a competition,” one of them chips in. Sure enough, portly hip-hop titan DJ Khaled will sign up one the winning act for his label.

Teamwork … Rebel Wilson and Anna Kendrick in Pitch Perfect 3. Photograph: Allstar/Universal Pictures

But things don’t go quite as expected. The Bellas’ tour rivals include a country band, a hip-hop outfit and an all-girl guitar band (brilliantly named Evermoist). When they throw down the riff-off gauntlet, the other bands prove to be better at everything – mashing up cover versions, writing their own songs and playing instruments. You know, like real musicians. At this point, Pitch Perfect’s entire raison d’être threatens to collapse like an overcooked soufflé, but this instalment realises the music was only ever really secondary to proceedings.

Those proceedings include the emergence of Fat Amy’s long-lost father (John Lithgow, whose attempt at an Australian accent might best be described as “transpacific”), Beca’s friendship with a handsome, sensitive young music producer, bouts of “we’re a second family” bonding, innumerable minor Bella sub-subplots, and the sniping Greek chorus of commentators Elizabeth Banks and John Michael Higgins, who are following the Bellas for a documentary. All of these play out through a whistlestop tour of European locations, costume changes and tightly choreographed song-and-dance numbers.

It’s a testament to writer Kay Cannon (who wrote all three Pitch Perfects) and new director Trish Sie that all these random elements stay in tune with each other and everything wraps up neatly. The comedy rarely falters. There are choice one-liners (“we’re going to be clinging to you like mom jeans to a camel toe”), music-industry satire (DJ Khaled travels with a dedicated “juiceologist”, with his own portable beehive), and Rebel Wilson’s offbeat interjections, which are judiciously kept in control, though she’s unleashed in the inevitable action finale like a secret weapon.

Permeating the nonsense, though, is a heartfelt assertion of teamwork and female solidarity in defiance of everyday sexism that’s very much in tune with the present moment (even if it’s best not to ponder too deeply what kind of “empowerment” the Bellas prancing in clingy camouflage outfits in front of US military hardware really represents). So many movies end with trite sentiments about “family” and “sisterhood” but it doesn’t feel forced here. It looks like these performers are genuinely enjoying themselves, and it’s infectious. Despite the preposterous escapades and the self-destructive satire, at the end of it all, there’s something real left standing.