Large numbers of riot police were deployed across the country for the rallies called by three opposition groups in defiance of government bans.
Tensions have been soaring in the country where Kabila, in power since his father's assassination in 2001, is widely thought to be preparing to remain in office beyond the two terms allowed under the constitution.
"A policewoman was killed by the crowd and a protester was also killed," in the eastern city of Goma, Jose Maria Ajanaz, who heads the United Nations Joint Human Rights Office (UNJHRO) in the deeply restive nation, told AFP.
Locals said Goma police fired tear gas to disperse youths who had set up barricades and blocked roads.
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Clashes also broke out in the north of Kinshasa, where about 5,000 demonstrators were on the streets to oppose moves to keep Kabila in power beyond the expiry of his second term in December.
Police said that while the Kinshasa demonstration had the go-ahead from authorities, the crowds had deviated from an agreed route.
"In these cases we don't negotiate, we disperse," national police spokesman Colonel Pierre Rombaut Mwanamputu told AFP.
Police fired tear gas against stone-throwing protesters and then charged them to break up the demonstration, AFP journalists said.
The protestors in Kinshasa, home to about 10 million people, shouted slogans such as "No to a third mandate", "People rise up" and "Kabila buzz off."
As well as fearing Kabila will seek to extend his rule, opposition groups are also angry that his powerful rival, football magnate Moise Katumbi, was all but pushed into exile in South Africa last Friday.
In the second city and mining hub of Lubumbashi, dozens of riot police stood guard in the city centre where many shops remained shuttered and the number of vehicles on the roads was far lower than usual, according to an AFP team.