A five-year-old and an eight-year-old were among the four people who died during their boat sank off Dunkirk in the north France mardi.
15 other migrants were rescued, including eight suffering from hypothermia and two in cardiac arrest, local firefighters said.
French authorities said further research was underway in the region and the death toll could rise.
The tragedy is believed to be the biggest loss of human life during the current migrant crisis, bringing the total number of deaths since 2018 to 10.
Clare Moseley, founder of the refugee charity Care4Calais, said the “horrific” incident should be a “wake up call” for British and French leaders.
“We mourn the victims, we stand in solidarity with their families and friends,” she said.
“It is cruel and horrible that this time young children are among the victims.”
She added: ‘We must provide a safe and legal process by which refugees can have their asylum claims heard in the UK, this is the way to end terrifying and dangerous sea crossings and to stop the tragedy. again. ”
The Save the Children charity called on London and Paris to “develop a joint plan to ensure the safety of vulnerable children and families”.
“La Manche must not become a cemetery for children,” he warned.
Prime Minister Boris Johnson described the incident as “terrible” and said the UK had offered “all its support” to the French authorities, while the Home Secretary Priti Patel said she was “really saddened”.
Ms Patel added: “We are in contact with our French counterparts who are leading the response and have offered all the support they need to investigate this incident.
“This tragic news highlights the dangers that come with crossing the English Channel and I will do everything possible to prevent callous criminals from exploiting vulnerable people. ”
French Citizenship Minister Marlene Schiappa said she was “deeply saddened” by the incident.
“Despite the resources of the State, which have all been mobilized in Operation SOS, the losses are heavy and the final outcome is still uncertain,” she said.
People on the migrant boat had attempted to reach the UK despite gusts of up to 18 mph.
After seeing the ship in difficulty, French patrol boats and a helicopter were dispatched, as well as civilian boats.
Fifteen people were taken to hospitals in Calais and Dunkirk, according to the maritime prefecture of the Channel and the North Sea.
An investigation is underway into the cause of the sinking.