At least 7,000 people have been living there in squalid conditions.
Migrants queued peacefully to be processed, and 40 of the 60 coaches that will carry them to migrant centres across France have now left.
There is concern that some people will refuse to go because they still want to get to the UK, and that weekend clashes with police could be repeated.
The dismantling of the camp is expected to start on Tuesday.
How Jungle clearance began
What next after the Jungle?
The UK has begun to accept some of the estimated 1,300 unaccompanied children from the camp but halted the transfer process on Monday at the request of the French.
Read more on this story:
How are child migrants’ ages checked?
The desperate children of the Calais Jungle
Migrant children dream of getting to UK – Lyse Doucet
The authorities in the camp are determining whether individual migrants are with family members or travelling alone, and whether they are deemed to be vulnerable.
After processing they will leave for various parts of France and be given the opportunity to either claim asylum or face deportation.
Media captionOne migrant at the Calais camp tells the BBC: “I will not move one inch”
Media captionOvernight unrest in the final hours of the ‘Jungle’
Officials have predicted that some 2,500 people will leave the camp on Monday.
They said they were pleased with the operation’s progress although the Calais police commissioner, Patrick Visser-Bourdon, said some people waiting might have to return for transfer on Tuesday because there might not be enough buses.
Some volunteers have complained that the migrants were not given enough information about the day’s plans and warned of a sense of confusion and chaos.
About 7,500 beds are being made available in 450 centres across France. So far migrants have been have been taken to the regions of:
‘My dream is dead’ – at the Jungle with the BBC’s Gavin Lee
Migrants at the JungleImage copyrightAP
Rue des Garennes links the Jungle camp to the new migrant processing centre, and is a street of suitcases and resignation.
At 05:00, three hours before the clearance operation was due to start, groups of refugees and migrants began to form a queue. Since then, a mass exodus from the camp and hundreds are now lining the road waiting for a coach to take them away.
Towards the back of the queue is Adil from Sudan, carrying two bags, a football and a guitar. “My dream is dead, the people you see here, they are broken. We can’t believe it’s over.”
Inside the camp, aid workers from Care for Calais are moving tent to tent, warning migrants that if they do not leave, they will be arrested. The police presence is large, with many huddled in riot vans, keeping out of the cold, and making the most of the calm.
Children will be housed in the camp’s converted shipping containers while the rest of the Jungle is dismantled.
From Tuesday, heavy machinery will be sent to clear the tents and shelters that have been left behind. The whole operation is expected to take three days.
The French interior ministry said it “does not want to use force but if there are migrants who refuse to leave, or NGOs who cause trouble, the police might be forced to intervene”.
There are reports that British activists from the No Borders group have travelled to the Jungle to try to disrupt the demolition process.
The Jungle has played host to scenes of both squalor and of violence, as migrants, mainly from Africa and the Middle East, attempt to board lorries bound for the UK, clashing with drivers and police in the process.
A UK-funded wall 1km (0.6 miles) long is being built along the main road to the port in an attempt to deter would-be stowaways. The UK government is reported to have contributed about £1.9m (€2.2m) to the cost of the wall, which is due to be finished by the end of the year.
Why do migrants want to come to the UK?
Most believe there is a better prospect of finding work. Many want to claim asylum, although others want to enter incognito and remain as illegal workers.
The language issue is also important – many speak English but do not have a European language. Some also have relatives in the UK and that is a big draw.
Some are attracted by a belief that there is better housing and education available.
Some commentators believe illegal migrants also perceive Britain as a “soft touch” for benefits and a better place to find “black economy” jobs, although studies do not necessarily back up this view.
Speaking to the BBC, the head of the organisation that runs the port of Calais, Jean-Marc Puissensseau, said business had declined due to attempts by migrants to board trucks bound for the UK.
“The port has been really suffering [for] more than one year, because each night there were some attacks, or [migrants] were throwing branches, trees, everything to try to slow down the traffic and then to get into the trucks,” he said.
He called for action to stop migrants returning after the clearance.
What is the Jungle?
A migrant walks through the Jungle camp in CalaisImage copyrightAFP
The Jungle camp is near the port of Calais and close to the 31-mile Channel Tunnel
Officially about 7,000 migrants live in the camp. The Help Refugees agency said the final population ahead of its demolition was 8,143
The camp was halved in area earlier this year but the population continued to rise, and reports of violence have increased
Many migrants attempt to hide themselves in cargo vehicles entering the Channel Tunnel
The area has been hit by protests from both locals and truck operators
Last year more than one million migrants – many fleeing the civil war in Syria – arrived in Europe. Countries struggled to cope and division arose in the EU over how best to deal with resettling people.
In March, the EU struck a deal with Turkey to try to stop migrants crossing from Turkey to Greece while Balkan nations closed their borders to migrants.
As a result, the number of arrivals using the so-called eastern Mediterranean route has fallen.
However, migrants from African countries such as Eritrea and Somalia as well as West African nations such as Nigeria and the Gambia are continuing to attempt the crossing from Libya to Italy.
Migrant arrivals via Mediterranean, 2016
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.
Are you in the area around Calais? Have you been affected by the clearing of the camp? Let us know about your experiences. Email email@example.com with your stories.
Please include a contact number if you are willing to speak to a BBC journalist. You can also contact us in the following ways: