BELGRADE, Serbia—The night in mid-March when protesters stormed the headquarters of Serbia’s public broadcaster began like many recent Saturday nights in the Serbian capital. Weekly protests against the government of President Aleksandar Vucic had entered their fourth month, and several thousand people turned out for a mile-long march across the city. They planned to vent their frustrations over escalating political violence and democratic backsliding in the country.

The previous 14 protests had largely unfolded without incident, and there was no reason to believe this one would be any different. But as protesters made a pit stop in front of Radio Television Serbia, which is widely regarded as a government propaganda service, events took an unexpectedly confrontational turn. ...