NEW DELHI—Before joining a protest last week over voting rights, Abhijeet Dipke, the founder of the Cockroach Janta Party youth movement, issued a public challenge to Indian Prime Minister Narendra Modi.

“I know that I can be picked up anytime tonight or tomorrow,” he said in a press conference. “I want to tell Narendra Modi…We aren’t afraid of this type of thing…come and get me.”The next day, police seized the 31-year-old, who was clad in a T-shirt bearing the image of Mahatma Gandhi and holding an Indian flag. Police also detained Neha Bora, a prominent left-wing activist, after arresting two other Cockroach Party leaders earlier in the week. After Dipke and Bora were released from custody, they made it clear they would keep protesting.

“Now just waiting for my brothers to be free,” Dipke said in a social-media post with a photo of other Cockroach Party members flashing victory signs while in police custody.

Over 12 years in power, Modi’s Hindu nationalist government has gained a reputation for being heavy-handed toward dissent. Rights groups say the frequent use of antiterrorism laws to detain political activists for years without bail, and many tax evasion and corruption investigations against political opponents and news outlets have created a chilling effect.

The government maintains it applies the law impartially to fight corruption and safeguard national security. But its tactics in recent days, including putting tens of thousands of police and paramilitary troops on the streets of India’s capital and repeatedly detaining protest leaders, have been met with a mix of wit and defiance by the protesters. This has left the Modi government struggling to respond.

“The government seemed to be scared of nothing and no one,” said Ashutosh Varshney, a professor of political science at Brown University. “Now it seems to be scared of young people.”The prime minister’s office didn’t immediately respond to a request for comment.

The Cockroach Party had already wrested one success from the government in July, when it staged protests over the management of competitive exams and lack of economic opportunity. Left-wing student movements, such as the one headed by Bora, joined in. The scale of those demonstrations caught the government off guard. The education minister resigned that month, meeting the protesters’ top demand.

Then came anger over a sweeping revision that India’s election authority has undertaken to the country’s electoral rolls, removing some 130 million names, according to tallies from official data by activists. That sparked fresh protests this month. Police hauled off thousands of young people on Oct. 10 as the government sought to thwart the demonstrations.

A video clip of a sprinting Bora dodging slow-moving police quickly went viral. Images of those confrontations helped make this month’s protests far more about the government’s ruling style—and about Modi himself—than earlier protests by farmers over agricultural policy changes, or demonstrations against citizenship law changes that appeared to favor Hindus while excluding Muslims.

In the past, young people—a huge share of India’s population—were seen as a key source of support for Modi and his Bharatiya Janata Party.

India’s next national polls aren’t until 2029, but early next year India’s largest state, Uttar Pradesh, is set to vote in local elections. In national elections in 2024, Modi’s party lost its majority in the lower house of Parliament largely due to losses in Uttar Pradesh, where youth anger over lack of jobs influenced the result.

“The youth are a very large proportion of the electorate,” said Varshney, the Brown University professor.

“The real danger here is electoral.”During the overhaul, electoral officials deleted tens of millions of people from draft rolls if they failed to return a form that was delivered to their last known address. Millions more were selected for scrutiny if they or their parents or grandparents couldn’t be found on previous rolls, or if a software program found spelling or age mismatches, or “logical discrepancies” such as families with six children or more.

Government ministers have said the exercise is meant to purge lists of deceased people or fraudulent voters, such as illegal immigrants. That has led people to fear that falling off the voter list could lead to other exclusions—such as from welfare programs—or even cast scrutiny over their citizenship itself.

The new government in the state of West Bengal has said people who aren’t on voter rolls won’t be eligible for several government welfare programs. Authorities there completed the revised voter lists earlier this year.

The scrutiny of family size as part of the voter revision sparked suspicion among many Muslims that the exercise sought to delete them from the rolls, given that Hindu nationalist politicians have made remarks about Muslims having larger families. The electoral authority and the government have denied that the voter revision process discriminates against Muslims.

But tens of millions of those removed are from India’s Hindu majority, leading to broader opposition this time than during past protests over citizenship policies that mainly affected Muslims.

“The government is taking away voting rights of so many Indians, not only Muslims,” said Nurshid Alam, a 19-year-old Muslim who was able to as a first-time voter recently while his mother, a two-decade voter, was struck from the rolls. “By taking away people’s voting rights, you are telling them that you are not citizens of this country.”Many also see a link between the voter-list purge and a campaign promise by Modi’s party to create a national citizenship, which, if carried out, could leave hundreds of millions scrambling to furnish the necessary documents to prove that their ancestors lived in the country. The government has announced no plans for such an exercise. The protests have already forced authorities to undo parts of the voter-list purge.

An election officer working on Sunday at a special camp to help voters said he and other local election workers received orders in recent days to process the paperwork of the voters flagged for “logical discrepancies” without requiring any additional verification documents. They also began using a form that didn’t have any requirement for people to provide voting records of parents and grandparents, the officer said.

“The recent youth protests have forced these changes,” the officer said. “They are having an impact on the ground for sure.”The Election Commission didn’t immediately respond to a request for comment.

Write to Tripti Lahiri at tripti.lahiri@wsj.com and Krishna Pokharel at krishna.pokharel@wsj.com