Eurostar has introduced an emergency timetable for the next week while the wintry conditions continue.
The operator said there were speed restrictions on services to and from London, Ebbsfleet and Ashford and passengers should expect delays.
Eurostar has also closed ticket sales until after Christmas Eve.
Passengers have been urged to cancel or postpone their journeys and not to travel unless it was "absolutely necessary".
At King's Cross St Pancras station in London, dense crowds packed the waiting area as trains were cancelled or delayed.
Scott Marquis was booked on a 1700 GMT train from Paris to Ashford on Sunday after spending the weekend at EuroDisney with his family.
He said: "I am still stranded now.
'Severe weather conditions'
"An awful lot of people were left with small children in the freezing cold. There has been a lot of tears and frustration and people losing their cool."
Eurostar has asked people not to travel to St Pancras on Monday even if they have a booking.
A Eurostar spokesman said: "Due to the severe disruption we have experienced today we have a number of trains and crews out of position.
"We expect ongoing severe weather conditions to result in speed restrictions and delays on our routes.
"We will operate a contingency timetable with some cancellations for a number of days.
"We recommend that if you do not need to travel you cancel or postpone your journey. Due to exceptional demand, sales are closed for travel up to and including 24 Dec."
Main roads across Kent are open, but motorists are urged to allow extra time for their journeys.
A Kent Police spokesman said: "Kent Police is advising motorists to take care as widespread icy conditions continue.
"Anyone who chooses to travel should make sure they are well prepared."
Southeastern trains and Southern Railway said a normal service was running on Monday, but there may be cancellations on some routes.
Robin Dodridge, the director of corporate operations at the Port of Dover, said the port was operating normally.
"The port is becoming well-versed in putting its plans into place for handling significant volumes of additional passengers when a major transport crisis hits the UK," he said.
Elsewhere, Kent Fire and Rescue Service said two nine-year-old boys had to be rescued in Herne Bay after walking across a frozen lake to an island.