More than 2,300 people were bussed away to reception centres on Monday.
Migrants had queued peacefully to be processed, but there are concerns some will refuse to go because they still want to get to the UK.
Some 7,000-8,000 people have been living there in squalid conditions.
The demolition of the camp is a sign from the French government to the people of Calais that their concerns are being taken seriously, says the BBC’s Jonny Dymond from the area.
It is also a message to remaining residents that the camp will not remain a place of shelter, our correspondent adds.
Christian Salome of the Auberge des Migrants charity said the process was working well so far because those leaving were the ones who wanted to.
But he added: “I’m much more concerned about later in the week when the only ones remaining are those who do not want to leave, who still want to reach England.”
There are warnings that those determined to stay will set up camp in the surrounding countryside while the demolition takes place before returning to the area.
Children are the only group allowed to stay in Calais. They will be housed in the camp’s converted shipping containers while the rest of the Jungle is dismantled.
Almost 200 children from the camp have been brought to the UK, some of them under the “Dubs” amendment to the Immigration Act, according to UK Home Secretary Amber Rudd.
The amendment allows particularly vulnerable children – such as girls and those under 13 – refuge in the UK, even if they do not have family already in the country.
But the process of transferring some of the estimated 1,300 unaccompanied children from the camp was halted on Monday at the request of the French.
The operation to clear the tents and shelters is expected to take three days. More than 1,200 police have been deployed to keep watch.
The French interior ministry said “police might be forced to intervene” if there was unrest during the demolition.
Long queues formed earlier outside reception centres in the camp as the French authorities worked to determine whether individual migrants were with family members or travelling alone, and whether they were deemed to be vulnerable.
After processing, they boarded buses to different parts of France where they will be given the opportunity to claim asylum, or face deportation.
