POLLS say immigration is voters’ main concern, so the issue was always going to play strongly in the Brexit debate. David Cameron’s Tory government has promised to bring net migration below 100,000 a year but the latest number was 362,000. Worse, much of the upsurge is accounted for by a rise in EU immigrants. That is why Mr Cameron fought so hard to win a four-year delay in granting in-work benefits to them in his EU renegotiation.

Despite Mr Cameron’s deal, immigration is one subject on which Leave campaigners have a clear lead. The correlation between hostility to immigration and support for Brexit is high, so if they can turn the vote into one about migration, they will win. Yet in trying to do this they not only ignore much economic evidence about the impact of migration but also muddle several unrelated strands of the subject.

They say Britain has lost control of its borders. In fact anyone entering Britain (except from Ireland) must pass through border checks. Or they point to Europe’s refugee mess, although since Britain is not in the EU’s Schengen passport-free zone, the country has largely escaped it. Some warn that the accession of Turkey will let in hordes of Turks. Yet Turkish membership is many years off and, if it were agreed, would come with tight migration limits. A few Eurosceptics use the terrorist attacks in Brussels on March 22nd to claim that free movement of people lets terrorists into Britain (the government says they show how vital co-operation on security is).

The Remain campaigners are not above their own scaremongering. Some suggest that Brexit might result in the 2m-odd Britons settled in Spain and elsewhere in the EU being sent home. This is unlikely, though questions might be raised over access to health care. Remainers have warned that France might scrap the Le Touquet treaty that places the Anglo-French border in Calais, bringing squalid refugee camps to Dover instead. Some French politicians might indeed want to end this unpopular deal, but it is a bilateral one and not linked to Britain’s EU membership.

The real argument should be over the effects of EU migration. It has certainly been bigger than expected. In 2003 one forecast said that up to 13,000 east Europeans a year would come; five times as many turned up. There are now about 3m EU migrants in Britain, the latest inrush from southern Europe. Yet over half of net immigration comes from outside the EU.

A post-Brexit Britain might not be able to stop EU migration anyway. If it wants to retain full access to the EU’s single market, it will probably be required to accept free movement of people, as Norway and Switzerland are (both have proportionately more EU migrants than Britain).

Most Brexiteers insist on tougher controls. They say heavy EU migration burdens taxpayers, drives up welfare spending, strains public services like health and education and aggravates the housing crisis. Some argue that migration steals jobs and reduces wages, especially for the lower paid. Those who favour some immigration often prefer an Australian-style points system that would let Britain cherry-pick the best and brightest.