Crowds surged towards the warehouse where processing was taking place as police opened the gates just after 7am UK time on Monday.
Caroline Gregory is a Christian and a long-term volunteer and communicator in the Jungle refugee camp.
She told Premier's Inspirational Breakfast that leaving refugees are making a special effort for the processing centres.
She said: "They've very much made the effort to arrive at these centres looking as dignified as possible, when you don't have much else to cling to, making sure that your shoes are clean and that you've had a hair cut from your friends is really important."
Ms Gregory went on: "These are humans and we should be caring from them as such, the Bible tells us specifically to look after the foreigner, the wanderer, the alien - it doesn't add but not if they're possibly over 18, or not if they've an economic migrant.
"I just believe that we should be looking after these vulnerable people as best we can."
Over the weekend there were violent clashes, with camp residents throwing stones at French riot police on the perimeter who fought back by firing tear gas.
Some 60 government-organised buses are expected to take thousands of the camp's residents to temporary reception centres where they will have to claim asylum in France within a set period of time or face deportation.
It is believed they will be sorted into groups of families, minors, vulnerable or ill people and others travelling alone at a registration centre in a warehouse by the camp and given coloured wristbands depending on which region they say they would like to be sent to.
Aid workers have been advising refugees and migrants to register for the buses together as they believe this will give certain groups of friends or communities the best chance of not being separated.
A further 85 buses are expected to arrive on Tuesday and Wednesday as the mass eviction continues.
Adam is has been living in the Jungle and told Premier's Rosie Wright why he is there.
He said: "In 2003 the war started in Sudan, no stopping, that is why were are here."
Mohammed is also from Sudan and told Rosie: "People are fighting between them, they are killing many people, people are dying, so that's why I'm leaving Sudan."
French President Francois Hollande announced last month that the camp will close before winter, with its estimated 6,500 inhabitants dispersed around the country.
Those who do not seek asylum will be deported.
Speaking on News Hour from queues of migrants which had formed at the Jungle, Premier's Rosie Wright said: "[French authorities] are taking their name, their age and their country of origin.
"Some of them in this queue know nothing about France. They don't know where they're going but they've decided this is a better option that what they've just left behind.
"What we're looking at now is partly quite calm because these are people who volunteered to be in the queue to register to get on the buses.
"The real drama will happen when the people who've refused to come out will start to be evacuated and that could start [as] soon as tomorrow."
Listen to Caroline Gregory on Premier's Inspirational Breakfast here:
Listen to Premier's Rosie Wright speak to Adam here:
Listen to Premier's Rosie Wright speak to Mohammed here:
Lister to Premier's Marcus Jones speak to Premier's Rosie Wright in Calais: