It's not hard to see why the Occupy Wall Street protests have gone global. What kicked off a month ago in relative obscurity – drawing inspiration from this year's Spanish indignados occupations and the uprisings in Egypt and Tunisia – has now spawned protests in more than 900 cities around the world. The only surprise is it didn't happen sooner.

Three years after the banks that brought the west's economies to their knees were bailed out with vast public funds, nothing has fundamentally changed. Profits and bonuses are booming for financial oligarchs and corporate giants, while most people are paying the price of their reckless speculation with falling living standards, cuts in public services and mounting unemployment.

Coming as this crisis has done – at the end of an era of rampant deregulation that has created huge disparities of income and wealth, concentrated in the hands of the top 1% and secured by politicians bought by corporate interests – a backlash against those actually responsible was well overdue.

The occupation slogan "We are the 99%" exactly reflects the reality in the crisis-hit Anglo-Saxon economies in particular – just as the protesters' call for systemic change has far stronger echoes in US public opinion than its captive political class would have anyone believe. A majority of Americans are sympathetic to the protests while a recent poll found only a narrow majority thought capitalism a better system than socialism – in a country where the term is as good as a political swearword.

That has now shaped the political and corporate response. While the protesters were originally ridiculed as unfocused, or denounced by leading Republicans as "mobs", they are now championed by the media establishment – including the New York Times and Financial Times – on both sides of the Atlantic. Obama has made friendly noises, while his officials say they now plan to "run against Wall Street" in next year's presidential campaign.

In a climate where plutocrats like Warren Buffett are meanwhile begging to pay higher taxes, it's a clear sign of elite anxiety at the extent of popular anger and an attempt to co-opt the movement before demands for more fundamental change get traction.

Something similar seems to be going on in Britain where – against a steady drumbeat of lobbying scandal and escalating unemployment – police and the conservative Daily Mail have so far both given the City occupation outside St Paul's Cathedral a notably easy ride.

Of course the London protesters, camped out in a tent city near the Stock Exchange, have also been abused as "muddle-headed" layabouts and "Toytown Trots". But despite their rejection of the current economic system as "unsustainable", their initial statement includes a call for "regulators to be genuinely independent of the industries they regulate" that wouldn't look out of place at a Liberal Democrat conference.

There's no doubt, though, that these occupations echo both the spirit and organisation of the anti-corporate movement that erupted in Seattle in 1999. The tactic of occupying a symbolic public space (as opposed to strikes, sit-ins and marches) can be traced back to Greenham Common in the 1980s through a string of often dubious "colour revolutions" over the past decade.

But it's this year's drama in Tahrir Square (acknowledged with an Egyptian flag at the London camp) that has given it such evocative power. And while the 1990s anti-capitalist globalisation protests took place at a time of boom and speculative frenzy, today's occupations are targeting a global capitalism in the deepest crisis.

Which is why they have such a clear sense of reflecting the common sense of the age. What both movements now and then also share is an intense commitment to direct democracy and the influence of an "autonomist" opposition to engagement with mainstream politics – seen as a central part of the problem, rather than any solution.

In that, of course, they're in tune with millions. But when it gets to the point of resisting making direct political demands at all – an issue of controversy this week among US protesters, with some arguing "the process is the message" – that would surely limit the protests' impact.

The Occupy movement has already changed the political climate in the US. Some commentators argue that's enough – and it's up to politicians and wonks to turn the theme of economic justice into policy. But that would be to hand the initiative to the very system the protesters reject – and limit the scope for making common cause with others resisting austerity and corporate greed.

Not only that, but any demands need to be a good deal more radical than "independent regulation" if they're to make sense of the call for fundamental change and action to tackle the crisis: democratic ownership and control of banks and utilities, say, and wealth and transactions taxes for a start.

And as Naomi Klein argued to protesters in New York, the movement will also need democratic structures and institutions if it's to put down roots rather than fizzle and burn out. Trade union support for the US protests is a promising sign, as is the London occupiers' backing for next month's pensions strike and yesterday's electricians' blockade of a Balfour Beatty construction site over threats to rip up contracts.

The form and focus of these protests already varies widely from country to country: in Chile, they originally concentrated on free education, but now the target has expanded to include banks and GM crops. Across Latin America, where the revolt against neoliberalism first began more than a decade ago, it has been alliances of social movements and political organisations that have proved most successful in turning protest into economic and social change.

But there is of course no automatic link between large-scale protest and any radical political breakthrough: Spain has been convulsed with occupations and strikes – and is expected to elect a rightwing neoliberal government in reaction to the socialist government's austerity. The populist right can take advantage of mass disaffection as well as the left.

But in just a few weeks the Occupy movement has helped bust open the political class veto on the scale of change demanded by the crisis – and now that opportunity needs to be seized.