Alarge proportion of the British artists who have played at the London jazz festival in the past 26 years can trace their ancestry back to the Windrush generation, so it seems fitting for the festival to commemorate the 70th anniversary of the Empire Windrush's arrival. Tonight’s event, curated by Trinidadian poet and novelist Anthony Joseph, is billed as “A Celebration” but it is an angrier and more militant show than that word implies.

It starts with joy – a bubbly version of the Harry J Allstars’ old ska classic The Liquidator, then a star turn from the winding-and-grinding 78-year-old carnival queen Calypso Rose – but one gets the impression that Joseph isn’t a natural celebrator. Dressed in a sharp suit and a trilby (looking not unlike one of the Windrush emigres disembarking at Tilbury docks in June 1948, footage of which is projected above the band) Joseph is much better at righteous anger, a register that rather suits the mood. The music gets darker, and the songs are accompanied by a series of suitably militant quotes by Caribbean writers (Earl Lovelace, Pearl Connor-Mogotsi, Sam Selvon, Kamau Brathwaite, Frantz Fanon) and footage of political protest.

Despite the attempts at a narrative thread, tonight’s gig often just works as a variety show, and there are plenty of great performances. The 10-piece band, led by Jason Yarde, are superb, blurring the boundaries between reggae, soca, Afrobeat and free jazz. Cellist, pianist and singer Ayanna Witter-Johnson brings the house down with her spartan reading of the Police’s Roxanne, accompanied only by the strums and slaps of her cello and the funereal harmonies of alto saxophonist Yarde, trumpeter Byron Wallen and trombonist Harry Brown. Brother Resistance's Ring de Bell is a tremendous piece of politicised soca which the band transform into a Fela Kuti-like anthem, while south Londoner Gaika performs several pieces of gothic dancehall, putting his voice through an eerie harmoniser. Cleveland Watkiss adds a righteous reggae skank to Syl Johnson’s southern soul song Is it because I'm black? and transforms Omar’s There’s Nothing Like This into a piece of lovers rock. Lord Kitchener's Windrush anthem London Is the Place for Me is performed twice: first as a jaunty instrumental, then later as a quizzical, mournful version in a minor key, sung by Joseph’s teenage daughter Meena.

Sometimes the overlap between militancy and entertainment is unintentionally hilarious: images of Black Lives Matter protests are still on screen as the ribald 83-year-old calypsonian Mighty Sparrow takes to the stage and sings his distinctly un-PC comedy song Congoman, about two white women who travel to Africa and get eaten by cannibals. At other times the mixture becomes uncomfortable. Precious, a jerky piece of drum’n’bass scat-sung by Watkiss, is accompanied by photographs of 36 black Britons who have died in police custody since 1971. It is an intensely powerful montage but it nearly kills the evening stone dead. That is followed by the centrepiece of the show, a 25-minute instrumental song-cycle entitled The Windrush Suite, but by this point it’s nearly 11pm and hundreds of audience members are discreetly slipping out. When Joseph brings the suite to a close, it is greeted by a kind of stunned, shocked silence.

Windrush: A Celebration was at least half an hour too long, and could have done with some more joy – it didn’t seem to locate anything to actually celebrate about the immigrant experience. But, as a way of marshalling seven decades of slow-burning rage into art, it was often devastatingly effective.