An crowd of about 3,000 forced their way through police barricades in their attempt to join a group of some 50 other families who had been staging a 2-day sit-in nearby.
Police used pepper spray and water cannons and traded punches and kicks with those at the front of the crowd.
"Salvage the Sewol ferry", the protestors chanted.
Earlier, police had pushed several protesters into police vans.
Others crawled under police buses to avoid arrest and one mother passed out, requiring help from an ambulance.
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The sinking of the vessel in April last year claimed 304 lives, most of them teenagers on a school journey.
On Thursday, mourners clashed with police in the same place after a mass rally drew 10,000 people to mark the first anniversary of the disaster.
A group of some 100 victims' relatives have been staging a sit-in protest on the pavement for the past two days.
President Park Geun-Hye, currently on a trip to South America, vowed on Thursday to raise the sunken ferry to the surface to help find the nine people who are still unaccounted for.
But the relatives were unappeased, saying Park had failed to give other assurances on their demand for a fully independent inquiry with no government interference.
While largely blamed on the ship's illegal redesign and overloading, the Sewol disaster laid bare deeper-rooted problems of corruption, lax safety standards and regulatory failings attributed to the country's relentless push for economic growth.
Public opinion has been largely supportive of the families, although some conservative groups say left-wing organisations have hijacked the cause in an effort to embarrass the government.