‘The Dance of the Pan Pan at the Monico’, a painting by Gino Severini

When futurist impresario Filippo Marinetti travelled to Paris and Moscow before the first world war proclaiming himself “the caffeine of Europe”, he was playing to art’s first global market. Everything the hyper-stimulated Marinetti loved about modern life – noise, machines, big cities and “a new beauty: the beauty of speed” – was shaping not only art on the canvas but also its international dialogue.

Between 1900 and 1914, train travel and the growth of the cosmopolitan metropolis brought artists together as never before. Picasso and Juan Gris from Spain became Parisian cubists. Kandinsky from Russia stirred up expressionism in Munich. Parisians Francis Picabia and Marcel Duchamp reinvented themselves as New York celebrities. Tate Modern’s new exhibition Futurism, marking the centenary of Marinetti’s 1909 manifesto promising to reject tradition with “courage, audacity and revolt”, is above all remarkable as a record of those heady exchanges. Without them, the great transformations of modern art could not have happened.

Apart from Picasso’s seminal bronze “Head of Fernande” (1909) – the first cubist sculpture: a faceted head, features jagged as the mountains of Horta de Ebro where the piece was conceived, with Fernande’s hair massed into thick prehistoric-looking volumetric curves – every work here was made between 1911, when the Paris Salon first showed cubism, and 1915, by when artists across Europe were repatriated and often enlisted in the war. Between these dates, futurism hit Paris with a display at the Galerie Bernheim, continuing at London’s Sackville Gallery. That show is, as far as possible, reconstructed at Tate, and enlivened by an international context.

A blazing room of Russian cubo-futurists includes, from St Petersburg, Alexandra Exter’s hurling urban fragments lit by an inner radiance in “City at Night” and Malevich’s “Aviator”, composed of truncated metallic cones, his headgear a black quadrilateral engraved with a zero, anticipating the “zero of form” in “Black Square”. There are English vorticists, French Orphists led by Delaunay and his dislocated “Eiffel Tower”, and examples of the early School of Paris as dazzling as Léger’s ethereal white-red tubes and cylinders in “Contrast of Forms” and as heavy-handed as “Teatime (Woman with a Teaspoon)” by Jean Metzinger, whom Picasso mocked as a “dreadful straggler” in cubism’s wake. It is a fabulous panorama of a historical moment, when every serious painter was negotiating his own way round Picasso’s and Braque’s experiments: still, near-monochrome cubist still lifes, carefully arranged in the atelier, that occupy a small room at the core of this exhibition and contrast, in their lofty sobriety, with everything else.

‘Unique Forms of Continuity in Space 1913’, cast 1972, by Umberto Boccioni

“Take down all possible information about the cubists. Go to Kahnweiler’s and see if he has any photos of recent works, buy one or two,” Umberto Boccioni in Milan wrote to Gino Severini in Paris in 1911. What distinguished the Italians is that they brought cubism out of the studio and on to the street, using its fractured language to depict massed crowds, surging movement, the rough and tumble of everyday urban existence, painted in garish, acrid colours, often at monumental scale.

Their largest works – such as Severini’s four-metre music hall fantasy “The Dance of the Pan Pan at the Monico”, Carlo Carrà’s drawn-out analysis of movement in “The Funeral of the Anarchist Galli” – aim to immerse the viewer in the picture’s action and today they look overblown and over-schematic, like jigsaw puzzles. At a more modest scale, futurism’s celebration of dynamic modern life is convincing and, in its passionate youthful optimism, affecting too: the marquetry of coloured planes and block-like figures in Severino’s “The Boulevard”; the arrhythmic forms and acute angles of Carrà’s “What The Tram Told Me”. In “The Rebellion”, Luigi Russolo uses a simple geometric vocabulary for his red mass of demonstrators, marching over the canvas like arrows. In Boccioni’s crystalline, whirling “The Forces of the Street”, a cubic tram duplicated in a network of zigzags illuminates a night scene as if by electric light.

Yet cubism, meditative and cerebral, could not alone deliver Marinetti’s vision of sound and fury in paint. For all their disdain for the past, the futurists crossed it with divisionism – the breaking down of light and colour into pointillist dots, derived from Signac and adopted by a school of late 19th-century Italians. Many of this show’s most engaging pieces, therefore, would not have looked out of place in the 1890s: the fleeting blue-mauve-red figures in Carrà’s melancholy nocturne “Leaving the Theatre”; Giacomo Balla’s “Girl Running on a Balcony”; “Modern Idol”, a grotesque reveller, spectacularly made-up and caught in a spotlight, who “shrieks out the most heart-rending expressions of colour”, according to Boccioni, as a symbol of riotous energy.

You can see why Kahnweiler, Picasso’s and Braque’s dealer, forbade his artists from exhibiting with the futurists because “they made too much noise”. The juxtaposition here with French cubism and Russian cubo-futurism, both of which had formal agendas – exploring limits of representation, possibilities of abstraction – highlights futurism’s limits. It explains why the movement was so brief, and also why its reverberations were not painterly but rippled out into Dada’s absurdities, constructivist experiments, and the chilling dissections of human emotion as mechanised acts by Picabia and Duchamp.

Futurism, with its glorification of mechanisation and war – “the world’s only hygiene”, according to Marinetti – did not survive the 1914-18 conflict and Boccioni, its most ambitious and assured artist, was killed in military training in 1916. How would he have developed? In his sculpture especially, where the dialogue with Picasso is charged, there are suggestions that a major 20th century talent was lost.

“It’s strange and terrible! I struggle with sculpture: I work, work and work and don’t know what I give. Is it interior? Is it exterior? Is it sensation? Is it delirium? Is it brain? Analysis? Synthesis? I don’t know what the f*** it is. We’re all a bunch of dickheads,” he wrote to Severini in 1912.

Out of this chaos came his striding man-machine “Unique Forms of Continuity in Space”, one of the greatest works in Tate’s collection. It is joined by “Antigrazioso” (“Anti-graceful”), a bronze portrait of Boccioni’s mother, her rough features winding like an uneven staircase – an answer to Picasso’s “Head of Fernande” – and “Development of a Bottle in Space”, which dares to assimilate a key cubist motif into a futurist icon of explosive movement. As the bottle, dissected and reconstructed, rolls spiral-like around an axis, it surges lyrically up into space and out into the future, like Tatlin’s constructivist tower: fervent, idealistic, doomed and, a century later, still compelling.

‘Futurism’, Tate Modern, London, to September 20. Tel: +44 (0)20 7887 8888