Police said 2,300 people demonstrated in Paris, organisers put the figure at 4-5,000, to protest at the assault on Théo Luhaka on a working-class housing estate in Aulnay-sous-Bois, near Paris.
Left-wing presidential candidate Jean-Luc Mélenchon, Green MP Noël Mamère and former footballer Lilian Thuram joined their ranks.
Thirteen people were arrested after several hundred people threw projectiles at police, who responded with teargas.
Some 350 people demonstrated in the western city of Rennes, where four arrests were made, and there were smaller marches in Nice, Le Mans, Poitiers, Montpellier, Dijon, Nantes, Angers, La Rochelle, Amiens and Avignon.
Right-wingers call for protest bans
Right-wing MP Eric Ciotti joined far-right presidential candidate Marine Le Pen in calling for the Paris demonstration to be banned as a threat to public order, saying it was about "police pseudo-violence".
On a visit to the south-western city of Perpignan, Interior Minister Bruno Le Roux was conciliatory towards the protesters.
"There must be calm in the police's work with total respect on both sides," he said. "This respect cannot be in just one direction."