Some officers fired shots in the air in the capital Kinshasa as church and opposition groups defied a ban by authorities to push ahead with the demonstrations.
The protesters were demanding that Kabila promise he will not seek to further extend his time in power in Democratic Republic of Congo, a mostly Catholic former Belgian colony.
Kabila has been in power since 2001. Elections to replace him have been delayed and are currently set for December 2018.
A churchgoer who asked not to be named told AFP that officers dispersed worshippers from a mass in the parish of St Michael's in central Kinshasa on Sunday morning.
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"While we were praying, the soldiers and the police entered the church compound and fired tear gas at the church," he said.
Another parishioner who identified herself as Chantal said: "People fell, first-aiders are resuscitating old ladies who have fallen" -- but added that the priest carried on saying mass.
Earlier at the Notre-Dame of Congo Cathedral in Gombe, north Kinshasa, security forces also fired tear gas as opposition leader Felix Tshisekedi arrived, AFP journalists saw.
The parish priest asked worshippers to "return to their homes in peace because there is a heavy presence of soldiers and police ready to fire".
In Kinshasa, Catholics of the "Lay Coordinating Committee" had invited worshippers to walk, holding bibles, rosaries and crucifixes, after mass on Sunday.
They want Kabila, 46, to declare publicly that he will not run for another term as president.
Kabila succeeded his assassinated father Laurent Kabila in 2001.
He refused to step down at the end of his second and final term in December 2016.
That refusal led to protests and a bloody crackdown.
Elections had been due to take place by the end of this year under a church-mediated deal.
The delayed poll is now scheduled for December 23 next year.
Congolese authorities cut off internet access "for reasons of state security" before the planned march on Sunday.
The army and the police deployed in large numbers at churches across Kinshasa, the capital of around 10 million people.
Government spokesman Lambert Mende alleged in televised comments that "weapons of war have been distributed" by opponents of the government.
"These destabilising acts of agitation aim to create an atmosphere of insurrection which would enable them to seize power in our country by undemocratic means," he said, citing a government report.
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