The student protests that have gripped India sent thousands of young people into the streets on Friday, where some were beaten by the police, as the government sought to buy time in negotiations with the movement’s leaders without conceding to their central demand.

In Mumbai and New Delhi, thousands of demonstrators turned out again to vent their anger over a botched medical school entrance examination that affected millions of young people, and a wider sense of frustration about a lack of jobs and opportunity. In Kolkata, police used batons to disperse crowds of thousands, according to local media reports, and there were demonstrations in Chennai, Hyderabad, Pune and Jaipur, as protests reached cities across India on Friday.

The youth-led protest movement is posing one of the most substantial challenges in years to the leadership of India’s prime minister, Narendra Modi. Many of the protesters are students, and others are working, but they share a sense of desperation and rage about education and employment prospects. Their anger is hard for Mr. Modi to ignore, with young people accounting for 40 percent of India’s population.

The self-described Cockroach Janta Party, which derived its name from an insult, is the driving force behind the protests. The group said on Friday that it would press for demonstrations nationally until Dharmendra Pradhan, India’s education minister, stepped down, a demand it has framed as nonnegotiable.

The group holds Mr. Pradhan directly accountable for the leak of the exams, which led to canceled results. But he is a key operative in Mr. Modi’s ruling Bharatiya Janata Party, credited with helping the party come to power in some recent state elections.

After talks with government ministers on Friday, Ashutosh Ranka, a spokesman for the protesters, said in a social media statement that the government had agreed “in principle” to compensate families affected by the suicides of students and would take “no legal action against protesters.” According to the C.J.P., at least 21 students have died by suicide since the exam leak, unable to handle the stress.

The government has requested more time to respond to the demand for the removal of the education minister, and another round of discussions is planned for Saturday.

Saurav Das, another spokesman for the protesters, said that, once they had achieved their main goal, the resignation of Mr. Pradhan, the group would also ask the government to consider its far more sweeping agenda for change, including judicial reforms, gender quotas for Parliament and punishments for politicians who switch political parties after being elected.

The prospect of an organized, student-led opposition to Mr. Modi, an acutely image-conscious leader, has added a new challenge to his recent days in office. He is also managing economic difficulties caused by the war in Iran and recently led his party to power in a series of state elections.

Some of the students’ fury is turning directly on him, but his response so far does not appear to have satisfied them. On Thursday, he posted on social media that “Nothing is more important than the welfare and future of our youth!” He has promised fast-track tribunals to punish those responsible for the exam leak. Even that acknowledgment was rare for Mr. Modi, a strongman leader, who must balance the need to placate the students with a desire to make sure that his political party does not look weak.

The protests in New Delhi and elsewhere have been going on for more than a month. But the protesters’ call to march to Parliament on July 20, its first day in session, has given the movement momentum. Thousands of people gathered in the capital for the march, and many have remained camped out at the protest site for days despite the authorities’ calls to disperse.

Anupreeta Das, Alex Travelli and Mujib Mashal contributed reporting.