Britain has warned it may take a "few days" to clear the backlog of around 4,000 lorries waiting to cross the Channel.
A man has been arrested after lorry drivers stranded by the French travel ban clashed with police in Kent.
More than 5,000 Heavy Goods Vehicles (HGVs) are being held in the southeast of England, with drivers being tested for Covid-19 before they are allowed into France.
There were reports of disturbances at Dover and at the lorry holding facility in Manston involving those waiting to cross the Channel this morning, Kent Police said.
Footage showed a handful of officers attempting to push back a crowd of protesting drivers in Dover.
One man has been arrested for obstructing a highway in Dover and remains in custody, Kent Police said.
Britain's Communities Secretary Robert Jenrick warned it may take a "few days" to clear the backlog of lorries waiting to cross into France, despite the travel restrictions being eased.
The British Retail Consortium (BRC) warned there may be shortages of some fresh goods until the backlog is cleared.
Andrew Opie, the BRC's director of food and sustainability, said: "Until the backlog is cleared and supply chains return to normal, we anticipate issues with the availability of some fresh goods."
France eased the travel ban today, but said those seeking entry into the country from the UK must have a negative coronavirus result.
All truck drivers, regardless of nationality, will be required to take a lateral flow test which can detect the new strain of Covid-19 and return results in about 30 minutes, the Road Haulage Association (RHA) said.
France imposed the travel ban in response to fears about the spread of the more infectious coronavirus strain, which is spreading in the UK.
Hauliers must be able to show proof of a negative test result carried out within the past 72 hours in order to be able to cross into France.
Those who return a positive coronavirus result will be offered a PCR test for a more "refined" result, Mr Jenrick told Sky News.
Drivers will then be moved into Covid-secure hotel accommodation to self-isolate for ten days if the second test comes back positive.
The disused airfield site at Manston has become the main testing centre for hauliers, with drivers required to self-administer the tests in their cabs under supervision.
There were 3,800 vehicles at the Manston site as of 9.30am today, along with 632 HGVs on the M20 as part of Operation Stack and 610 in Operation Brock, the Department for Transport (DfT) said.
"As remaining capacity at Manston is being reserved for testing facilities, any new lorries that are arriving are being directed to Operation Brock," a DfT spokeswoman added.
Hauliers have been urged not to travel to Kent until further notice.
Lorry drivers at the entrance of the Port of Dover's Eastern Docks have been honking their horns and shouting in protest at being stuck at the border.
Drivers standing in small groups shouted "we want to go home" as they tried to shelter from bursts of heavy rain and blustery conditions.
Shortly after 11am, police officers told drivers to move their vehicles away from the roundabout at the port's entrance to enable testing facilities to be set up.
RHA chief executive Richard Burnett said: "The massive backlog of vehicles will take time to clear and hundreds of drivers run the risk of not getting home in time for Christmas."
The Associated Press reported seeing a ferry pull into the French port of Calais before dawn, while trains carrying freight and car passengers were allowed to cross under the English Channel.
Roads into the Port of Dover remained gridlocked with vans and lorries this morning, with groups of people walking among the vehicles.
Transport Secretary Grant Shapps announced a temporary relaxation of drivers' hours for hauliers, increasing the driving limit of nine hours to 11, to help them get through UK borders safely over the coming weeks.
He said lorry drivers stranded in Kent by the French travel ban must show "patience" as work continues to clear the backlog.
Mr Shapps told the BBC: "The issue is just the logistics of people following the instructions and making sure we can keep the port clear in order that we get the traffic rolling.
"The more people that follow the clear instructions, the faster we can get this resolved."
Mr Shapps said it will take "a matter of days rather than weeks" but there will be "some patience required".
He added that the government is providing "welfare" for the lorry drivers stuck at the border, and will continue to do so in the days to come.
Irish hauliers warn of staffing issues
Separately, the President of the Irish Road Haulage Association has warned of staffing issues in the New Year due to Brexit and Covid-19 travel restrictions between the UK and France.
Speaking on RTÉ's Today with Claire Byrne, Eugene Drennan said that many drivers from Europe and Ireland will turn down work that involves travelling through the UK, due to fears of being caught in delays.
He said drivers are already a "scarce resource".
Mr Drennan called for Covid-19 testing to be provided in Ireland, for Irish lorry drivers, in order to expedite their journey through the landbridge and on to France.
He said that Ireland has the facilities to do this and he has written to the Government about the issue.
The World Health Organization's European chief Hans Kluge said limiting travel to contain the spread of the new variant was "prudent" until there was more information, but supply chains for "essential goods" and essential travel "should remain possible".
The European Commission yesterday recommended that EU member states lift the blanket bans some had imposed on arrivals from Britain to allow essential journeys to resume.
