Ferries are stuck at Dover and part of the M20 has been closed as lorries wait to cross the Channel.
The action by MyFerryLink workers has also caused Eurostar services to be suspended for the day, after protesters got on to the tracks.
Migrant activity is reported to have increased, with people attempting to board Kent-bound lorries in France.
The Foreign Office has warned travellers to keep their car doors locked in slow moving traffic heading back to the UK.
A spokeswoman for the Prime Minister David Cameron said he was being "kept updated" on the situation.
P&O Ferries said the MyFerryLink workers feared for their jobs after their ships were sold by Eurotunnel to DFDS at short notice.
P&O Chief Executive Helen Deeble said the sale had taken place with neither party appearing to engage constructively with French staff.
She also said her company was bearing the brunt of the disruption.
DFDS said it had received confirmation on Monday that Eurotunnel had accepted its offer to buy two ferries.
Carsten Jensen, senior vice-president at DFDS Seaways, said: "I understand that this is a difficult time for MyFerryLink employees but this is not a good situation for anyone and we hope this action will end swiftly."
Eurotunnel has not yet responded to the criticism by P&O.
In Kent, cross-Channel freight traffic waiting to cross Dover has built up and Operation Stack has been brought in to close part of the M20 so lorries can queue on the motorway.
Despite the strike being due to end at 20:00 local time (19:00 BST), Operation Stack will continue throughout the night and into the morning.
Dover Harbour Board said the industrial action closed the Port of Calais in the early hours and all ferries to and from Dover were affected, apart from DFDS Seaways services to Dunkirk.
Kent Police has said extra Dunkirk crossings have been arranged.
Tensions over illegal migrants at Calais have been rising, with the deputy mayor of the town blaming the British government for the number of people trying to board lorries.
Philippe Mignonet told BBC Radio Kent: "We will block the port - as simple as that. We'll arrange to block the tunnel if nothing is done".
In response, a Home Office spokesman said: "Law and order in and around Calais is the responsibility of the French authorities, but the UK continues to work with its French counterparts to strengthen the security of the border to stop illegal immigrants entering the UK."
Lorry driver Chris Cary, from Rochester, said he tried to cross the Channel back to Kent early on Tuesday but then decided to attempt getting on a train instead.
But he said he found the road on fire as he approached the terminal with about 500-600 trucks waiting to cross.
"I didn't cancel my booking this morning for P&O, so I've literally just come back to the port and I'm now sitting back on the lane where I was at 03:10 this morning waiting for a boat," he said.
Freight Transport Association's (FTA) international manager, Donald Armour, said blockades had brought the main motorway near Calais to a halt.
He said the FTA was very worried about someone getting hurt with the worsening migrant situation there.
"There is a lot of fighting between the migrants who all want to be on the best part of the road to get on to the lorries," he said.
"We haven't had a fatality but it's not good. The situation is definitely worse than it has ever been."
Truckers have been being advised not to stop within about 60 miles (97km) of Calais, to stick with other drivers and keep padlocks on their vehicles.
Dover port has urged people travelling with DFDS Seaways, P&O and MyFerryLink to check with their operators before travelling.
MyFerryLink has said its operations will end after 2 July and all bookings after that date will be refunded. The ferry operator has not yet commented on Tuesday's strike action.
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The Port of Calais is closed following blockades by striking French workers, causing severe disruption in Kent.