As thousands of people joined student rallies in Manchester and London today to protest against public spending cuts and the rise in tuition fees, the National Union of Students leader Aaron Porter had to be escorted by police away from angry crowds calling for his resignation.

Some of the protestors in Manchester turned on Porter – who had been due to speak at a rally in the city – calling him a "Tory too". Porter had previously been calling for unity in the student movement, which has fractured as opinions differ over how best to conduct the demos and sit-ins being organised around the country against the cuts and fee increases.

Eggs and oranges were also thrown by a handful of the protesters at Shane Chowen, the NUS vice-president, when he tried to address the crowd. Up to 5,000 people had gathered to hear speeches from trade union leaders and later some scuffles broke out between the police and a group of about 150 people who tried to force their way into the University of Manchester's student union.

In London the protest remained mostly peaceful, though an attempt by a group to break through police lines and reach the Tory party HQ at Millbank – where angry scenes took place at previous demonstrations – resulted in a handful of arrests.

In their march through central London to Parliament, the protesters chanted slogans including: "No ifs, no buts, no education cuts" and banged drums.

Anger at government proposals to raise university tuition fees to £9,000 from next year and scrap the Education Maintenance Allowance were the main focus of the slogans and placards. One drew an analogy between events in north Africa and the UK and read: "Ben Ali, Mubarak ... Cameron, you are next."

At a potential flashpoint along the route – Topshop in the Strand – students stopped to yell abuse directed at owner Philip Green, whose controversial tax arrangements have attracted fury.

"Pay your tax, pay your tax," they chanted.

The store was guarded by a line of police, keeping protesters apart from the bemused shoppers trapped inside.

Police handed out leaflets in London informing protesters about their right to demonstrate and what to do if trouble flared.

Moritz Kaiser, a 17-year-old sixth former from Oxford, was among those protesting.

"The tuition fee hike will affect my family quite badly and it is unnecessary when you look at how much is lost in tax avoidance," he said.

A dual British-German national, he now intends to head to the continent to avoid the additional bill.

"I was going to study here, but in Germany it is only ¤500 a year, and you get a free bus pass," he added.

His friend Lucio Pezzella, also 17 and at a sixth form college in Oxford, said the "wrong people were being punished" for the economic plight the UK finds itself in.

"Ordinary people shouldn't have to pay for a crisis brought on by the bankers," he said.

The general secretary of the UCU lecturers' union, Sally Hunt, addressing the rally in Manchester, accused the government of being at "war with young people".

"It is betraying an entire generation," she said.