Outside Downing Street, in front of a line of riot police, I am sitting beside a makeshift campfire. It's cold, and the schoolchildren who have skipped classes gather around as a student with a three-string guitar strikes up the chords to Tracy Chapman's Talkin Bout a Revolution. The kids start to sing, sweet and off-key, an apocalyptic choir knotted around a small bright circle of warmth and energy. "Finally the tables are starting to turn," they sing, the sound of their voices drowning out the drone of helicopters and the screams from the edge of the kettle. "Finally the tables are starting to turn."

Then a cop smashes into the circle. The police shove us out of the way and the camp evaporates in a hiss of smoke, forcing us forward. Not all of us know how we got here, but we're being crammed in with brutal efficiency: the press of bodies is vice-tight and still the cops are screaming at us to move forward. Beside me, a schoolgirl is crying. She is just 14.

"We followed the crowd," she says. So did we all. There are no leaders here: the thousands of schoolchildren and young people who streamed into Whitehall three hours ago in protest at the government's attacks on further and higher education were working completely off script. A wordless cry went up somewhere in the crowd and they were off, moving as one, with no instructions, towards parliament.

But just because there are no leaders here doesn't mean there is no purpose. These kids – and most of them are just kids, with no experience of direct action, who walked simultaneously out of lessons across the country just before morning break – want to be heard. "Our votes don't count," says one nice young man in a school tie. The diversity of the protest is extraordinary: white, black and Asian, rich and poor. Uniformed state-school girls in too-short skirts pose by a plundered police van as their friends take pictures, while behind them a boy in a mask holds a placard reading "Burn Eton".

"We can't even vote yet," says Leyla, 14. "So what can we do? Are we meant to just sit back while they destroy our future and stop us going to university? I wanted to go to art school, I can't even afford A-levels now without EMA [education maintenance allowance]".

I ask her who she thinks is in charge. Her friend, a young boy in a hoodie, grins at me, gesturing to the front of the kettle, where children are screaming "shame on you" and throwing themselves under the police batons. "Us," he says.

This is a leaderless protest with no agenda but justice: it is a new children's crusade, epic and tragic. More fires are lit as the children try to keep warm: they are burning placards and pages from their school planners. A sign saying "Dumbledore would not stand for this shit!" goes up in flames.

This is also an organic movement: unlike previous demos, there are no socialist organisers leading the way, no party flags to rally behind. The word spread through Twitter and Facebook; rumours passed around classrooms and meeting halls: get to Westminster, show them your anger.

Suddenly, there is a rush from the front and the sound of yelling police as hundreds of protesters run back from the lines, frightened. "Don't throw anything!" implores a young, bearded protester with a megaphone. "Protect your friends – don't give them the excuse!" But no one is listening. Sticks are being thrown: the mood is enraged as people see their friends struck back or struck down. "Tory scum!" they yell. "I wish they weren't breaking things," says Leyla, "but this is what happens when they ruin people's futures."