The attack was launched by an unidentified assailant moments after 41-year-old prime minister, a former soldier who took office in April, finished his speech to tens of thousands of people gathered in Meskel Square in the center of Addis Ababa.
Protest organisers said at least one demonstrator was killed and several were wounded by gunfire.
“Even if I go back to my village, I do not know how to live anymore. I have lost all hope.”
The kidnapping is one of the largest since the jihadist group Boko Haram abducted more than 270 schoolgirls from the northeastern town of Chibok in 2014.
"I have to drive for three hours to work just because of my sexuality"
Former president Robert Mugabe stepped down following a military intervention in November.
Al Qaeda-linked al Shabaab militants claimed responsibility for the attack.
"My wife is crying daily. They are persecuting her… What am I without my wife and family? We are not safe."
Refugees "armed with sharp objects began to assault police offers" which led to tear gas being dispersed, says Rwanda's police.
Political foes and allies alike described Tsvangirai as a brave politician who challenged the status quo when few could dare stand up to Robert Mugabe.