There was also an overnight arson attack on the Brittany residence of the National Assembly head - though no immediate link was made to the actions against President Emmanuel Macron.
The "yellow vest" demonstrators, named for high-visibility car jackets, began in mid-November over fuel taxes then broadened into a more general revolt against a political class they view as out of touch with common people.
In Paris, several thousand marched on Saturday beside symbols of power such as the National Assembly and Senate.
Though mainly peaceful, some protesters threw objects at security forces, a scooter and a police van were set on fire, and some shop windows were smashed.
One participant's hand was severely injured when he tried to pick up a so-called "sting-ball grenade" used by police to disperse crowds with teargas, a police source told Reuters.
Another man had blood streaming down his face in front of a line of riot police.
The Interior Ministry put the total number of protesters around France at 12,000, including 4,000 in Paris. The police source, however, said numbers were higher, with 21,000 demonstrators taking part in rallies outside Paris.