We hear this argument a lot, but nobody in the Leave camp has ever been in favour of making it easier for non-EU people to migrate to the UK. A large proportion of non-EU migrants are, bear in mind, brown, Muslim and from non-Anglophone countries. Exactly the same people who normally want to pull up the drawbridge against those are trying to get out of the EU.
This is criticising the vote Leave people. rather than debating the arguments.
And it is not quite true.
(1) Many euro-sceptics report that their non white constituents complain
that it is exceedingly difficult to get their relatives visas for family visits.
(2) My wife is from the Philippines, and brown.
David Cameron can not do anything about EU migration but he can and
is making it harder for skilled nurses and doctors to immigrate.
She advises me that some of the most skilled who have no family
connections here have got fed up with the rules being changed,
and have decided not to stay here but go to Canada or the USA instead.
There's nothing about the EU that makes it more difficult to liberalise those immigration laws. The argument sounds nice, but it really doesn't have anything to stand on.
However if we vote Remain, the rules for external admissions to the EU.
will likely all be harmonised.
As indicated before, Voting Remain is not a no change option; it will almost
certainly be interpreted by the EU federalists as active consent by the UK
population for further federalisation and harmonisation across the board.
After all if it is to be one EU super-state without internal frontiers,
there will not be any practical infrastructure to stop a non EU migrant
from applying to the member state with the most lenient admissions and
then moving to another member state with better work or welfare prospects.