"Net migration into the UK form other pats of the EU is estimated to have been 190 000 (see here)in 2016 A similar number comes from the rest of the world, over which, with the exception of refugees and asylum seekers, for whom we have other international treaty obligations, the sovereign British government has total control.
Of those from the rest of the world, nearly half (actually 47%) are students. (see again above link) For anyone thinking rationally it is absolutely crazy to try to reduce the number of these (though our sovereign government is doing so). They bring desperately needed foreign currencies in fees and for living expenses, they keep our world class universities in funds, and by and large they gain a favourable impression of the UK which increases our international standing.
An additional 21% of this group have definite job offers, and where our NHS and care services would be without them Lord only knows.
Now to look at immigration from the rest of the EU.
There is much concern about the influx of job-seekers from the Eastern European countries which joined the EU in 2004. But EU reputations permitted transnational restrictions on immigrants from these countries for up to seven years. The sovereign governments of Germany and Austria imposed the restriction for the full seven years, the sovereign government of France for five years, others for shorter periods. Only the sovereign governments of the UK, Ireland and Sweden imposed no restrictions.
So for better or for worse, the profusion of Polish plumbers et al was the direct responsibility of the UK government, not the EU."
Unfortunately, Labour peers took a different view and decided to side with the government on this issue.