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 Dhamendra 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 suicide of students and would take “no legal action against protesters.”