Around 2,318 refugees, including 400 minors, moved out of the "Jungle" camp in Calais to reception and orientation centres as part of France's clearance plan.
France began clearing the "Jungle" camp in Calais on Monday as many refugees who have camped for months or years gave up on their dreams of reaching Britain.
As the government says, the camp which was home to 6,500 migrants, is closing on humanitarian grounds. The refugees will be relocated to 450 centres across France.
Following outbreaks of unrest overnight, the refugees chose instead with calm resignation to be relocated in France while their asylum requests are considered.
By the evening, 2,318 camp residents had been rehoused at reception centres across the country, Interior Minister Bernard Cazeneuve said.
Of these, "1,918 adults left Calais on board 45 buses to go to 80 reception and orientation centres situated in 11 regions of France," Cazeneuve further said.
Four hundred minors have also been "taken to provisional reception centres" within the camp ahead of their transfer elsewhere, he added.
French officials celebrated the peaceful start to the attempt to dismantle the camp, which has become a symbol of Europe's failure to respond to the migration crisis as member states squabble over who should take in those fleeing war and poverty.
The refugees from countries such as Afghanistan, Syria and Eritrea had wanted to reach Britain or join relatives already there and most had planned to seek work.
Some refugees said they were reluctant to leave. "My brother is in London. I want to go and live with him," said Khaled Oryakhil, in his 20s, from Kabul. However, he added: "I don't have any choice. I will go on the bus if they force us.
An important issue however is the the fate of 1,300 unaccompanied child refugees.
Cazeneuve said all unaccompanied minors "with proven family links in Great Britain" would eventually be transferred from the Jungle across the Channel.
His British counterpart Amber Rudd said 800 children from the camp had already been interviewed.
"We hope to reach the figure of a few hundred more over the next two to three weeks while the camp is being cleared," she told parliament in London. "Then we will have done our commitment to the French which we hope will be (to take in) approximately half of the children who were there."
About 60 buses left the camp on Monday and the government predicts the evacuation will take at least a week.
Despite the calm, charity workers expect hundreds will try to stay and cautioned that the mood could change later in the week when work begins on razing the camp.
"There's a risk that tensions increase in the week because at some point the bulldozers are going to have to come in," said Fabrice Durieux from the charity Salam.
Others warned that many refugees who remained determined to reach Britain would simply scatter into the surrounding countryside, only to regroup in Calais at a later date.
"Each time they dismantle part of the camp it's the same thing. You're going to see them go into hiding and then come back. The battles will continue," said Christian Salome, president of non-profit group Auberge des Migrants.
On Tuesday, demolition crews will move in to start tearing down the slum, one of the biggest in Europe where an estimated 6,000 to 8,000 people have been living in dire conditions.
The operation is set to continue through Wednesday.