A young woman stands in front of a blue police van in a rain-soaked hoodie, one hand pressed against the metal. Behind the windshield sit police officers.

The movement, which began as a satirical protest, has evolved into a nationwide challenge to Prime Minister Narendra Modi's government.

Supporters of India's youth-led Cockroach movement vowed Tuesday to continue their protest in New Delhi, a day after police used tear gas and batons...