At the very end of Inua Ellams' roving play about black masculinity, a young actor nips in for a last-minute trim ahead of an important audition. "What's the part?" his barber asks. "A black man," he replies. "A strong black man."

It lands a laugh because Ellams has spent the last 100 minutes obliterating that very notion. Barber Shop Chronicles presents such a panoply of black masculinity that the very idea of bundling black men into a nondescript catch-all seems entirely reductive. Introducing us to 30 individual characters in barber shops the world over, Ellams reminds us of the sheer cultural diversity wrapped up in blackness.

Barber shops are pivotal points in black communities – places where groups of men gather, groom one another and pass time together. "This is our pub," insists one regular. Simply by putting such a culturally exclusive space on a public stage, Ellams is doing something potent – opening up a closed space, airing its conversations on a wider scale and inviting others in.

In the Three Kings barbers in London, owned by Cyril Nri's old-hand Emmanuel, all sorts of accents jostle for attention: booming Nigerian and elastic Jamaican sit alongside the clipped south London of the previous owner's son Samuel (Fisayo Akinade). All of Africa is represented here and this is, in more ways than one, a play about roots. London's black community is made up of different diaspora – each with its own heritage, parentage, politics and perspective.

Ellams follows those threads back to Africa, popping into barber shops across the continent in individual scenes. In each, the same small ritual recurs: the swish of a cloak around a customer's chest, the brush of hair off the neck, the cleaning of clippers. The same sense of community, the same care for appearances, the same conversations and jokes and concerns. The same football match comes over the airwaves – Chelsea v Barcelona, the Champions League Final. The telephone cables spooling over Rae Smith's set are a mark of connectivity, even as Ellams highlights the specifics too. In every scene, there's local colour and politics – in the Mugabe apologist in Harare and the defiant suburbanite in South Africa, the Londoners debating languages and the Ugandans talking tribal differences.

Gradually, Ellams lets an argument accumulate out of these disparate conversations. Fathers recur, as do public role models – rappers and politicians – and the idea of inherited values, identities and histories comes to the fore. Ellams ties unresolved issues and historic injustices to the prevalence of mental health issues in black men, suggesting that the political affects the personal.

This is, however, a play that makes its points through acting, and a dynamic ensemble swap characters with relish – all these individuals with their own mannerisms and tics, their own styles and modes of speech. It's a showcase of sorts, but one that sets a new bar, demanding that black characters be as rich and as rooted, as specific and as rounded, as those on show here.

That's partly what makes it so moving; the cast have skin in the game, and Bijan Sheibani's production draws its energy from that. For all the fine, comic individual performances – Hammed Animashaun and Peter Bankolé providing real comic flair, Nri and Akinade instilling its emotional heart – the actors are absolutely an ensemble, harmonising through Michael Henry's African vocal arrangements and jumping into Aline David's explosive, expressive choreography.

Barber Shop Chronicles runs at the National Theatre until 8 July, then tours.


Watch our interview with Inua Ellams here