Border Controls at Calais

British border checks in Calais were set up in 2003 under the Le Touquet Treaty..

This means that at Calais and Dunkirk, passengers go through French exit checks as well as UK immigration entry checks before embarkation.

Before the Referendum:

If Britain leaves Europe, British Border Controls at Calais will go.

EU referendum: Hollande warns of UK exit ‘consequences’  3 March 2016

French President Francois Hollande has said he wants the UK to stay in the EU – and warned of the “consequences” for immigration and the economy of leaving.
A French government minister earlier suggested his country could end UK border controls in Calais.

“If Britain leaves Europe, right away the border will leave Calais and go to Dover,” Bertrand said last month. “We will not continue to guard the border for Britain if Britain is no longer in the European Union.
The days when the French handled the whole burden and the British signed a cheque are over”.
https://www.theguardian.com/uk-news/2016/mar/03/calais-border-treaty-brexit-what-is-france-saying

Asked whether he would do this, Mr Hollande said: “I don’t want to scare you, I just want to say the truth – there will be consequences.”

The British prime minister, David Cameron, has seized on the French misgivings as a way of boosting support for the remain campaign, warning last month that in the event of Brexit, France could tear up the Le Touquet accord, leading to migrant camps springing up in south-east England.

Leave campaigners have dismissed the Calais claims as “scaremongering”.
http://www.bbc.com/news/uk-politics-eu-referendum-35712463

The Remain campaign claimed that a Brexit will see the border controls returning to the UK and that there will be Calais-like jungle camps for illegal immigrants in Dover.

After the Referendum:

France agrees to keep Britain’s border controls in Calais.

Prime Minister Theresa May and French President Francois Hollande have reached a “very clear” agreement to keep Britain’s border controls in Calais, dispelling fears that UK’s decision to leave the European Union will trigger moves by France to push the border controls back to British soil.

French President Hollande added that the agreement is useful to both countries. “Le Touquet ensures that we can say to migrants that there is no point coming to Calais as they will not be able to cross it.”
http://www.bbc.com/news/blogs-the-papers-36862682