Images from the huge station showed scores of police trying to hold back protesters at the entrance.
Superintendent Mark Cleland, from British Transport Police, said officers will use full investigative resources to track down offenders.
Services have been suspended at the station as campaigners are blocking the tracks.
Hours later, King's Cross was closed after dozens of protesters gathered outside.
Red flags displaying the emblem of the proscribed PKK are also visible in images of the Manchester Piccadilly protest posted to social media.
Woodburn students to join national walkout
However, the law empowers schools to establish limitations to maintain order and minimize disruption to the school day. Hoffman said students would leave class on March 14 and march on Broadway to Chelsea City Hall for a gathering.
A spokesman for the Manchester branch of the Friends of Kurdistan solidarity group said about 400 people had gathered in the city for an global day of action in defence of Afrin. The action appeared to be in opposition to Turkey's war with the Syrian Kurds.
"This is an global day of protest to break the silence of the worldwide community which hasn't put any pressure on Turkey to stop what it is doing", said the spokesman.
Passengers leaving or returning to South Yorkshire via Manchester are experiencing extensive delays this afternoon, after protesters climbed onto the train tracks and blocked all rail lines through the station.
"This is a spot demonstration of angry people who feel the United Kingdom government has been quiet on this issue", the spokesman said.
"We will continue to maintain a police presence at this station and at other stations across the rail network", he added.