Immigration minister Damian Green last week said Britain did not need any more unskilled workers or family dependents; instead it should accept only migrants of substance. "We want only the brightest, top of the range professionals" who will "add to the quality of life".

As chance would have it, his remarks coincided with the opening of a new exhibition at London's Tate Britain designed explicitly to showcase the profound impact migrant artists have had on the native tradition. The show traces the way whole genres that seem typically British – landscape painting, for instance – were brought here by migrant painters before being naturalised.

The only surprise is that anyone should be surprised. A similar case could be made for literature, architecture, music and sport – as well as industry and finance. But a Conservative minister seeking to raise the barrier to entry even higher than it is at present should pause for thought. Firstly, on a straightforward humanitarian issue, it is neither easy nor likable for a government to want to bring in a Wolfgang Mozart while refusing his wife leave to remain. Nor is it high-minded to wish to lure highly qualified people – doctors and engineers – away from their home countries. It was said not long ago that there were more medical practitioners from Malawi in the Manchester area than there were in Malawi. It may not be wise or right to sponsor brain drains on this scale.

The second objection is pragmatic. Just as governments are not good at predicting the exact week in which crops will ripen – and therefore when short-term seasonal workers are needed – so they are fallible when it comes to predicting which migrants should qualify. One lesson in the story of migration is that it is not a uniform experience.

Who would have guessed that an uncouth 17-year-old boy called Michael Marks, a Jewish refugee from Poland, would amount to much when he landed in Hull in 1882? He didn't speak English – often seen as a fundamental qualification. But he set up a network of market stalls that grew, eventually, into Marks & Spencer, and stitched himself – as St Michael – into the fabric of the nation.

Two of David Cameron's most significant supporters are themselves children of migration. Steve Hilton, director of strategy, is the son of Hungarian refugees who came to Britain in 1956, and it makes sense that his biography led him to the right-of-centre, since his family brought an understandable distaste for totalitarian politics. Their name was Hircksac – they took their new one from the first hotel they stayed in, perhaps unaware that Conrad Hilton himself was Norwegian-American.

And Daniel Finkelstein – Times columnist, an adviser to William Hague and one-time Conservative candidate – comes from a similar background. His Jewish family fled Poland and national socialism. He once said that Thatcher made the Tory party more immigrant-friendly by making it more meritocratic.

If both men promote ideas and values that seem "typically" British, they are mirrored at the head of the Labour party by the Miliband brothers, whose grandparents left Jewish Warsaw and whose father, Adolphe, escaped the Nazi invasion of Belgium in 1944. The fact that Britain was his salvation did not, however, inspire in Ralph Miliband anything like political gratitude: he was a fiery leftwing lecturer at the London School of Economics and a socialist visionary. His sons, if anything, are Miliband-lite.

It is hard, then, to draw firm lines between migrant experience and political belief, since migration does not have predictable outcomes.

For every Peter Hain, a liberal agitator whose anti-apartheid sensibility came fully formed, there's a Michael Portillo, whose father was a pacifist professor and Republican exile from Franco's Spain, but who followed a different star in his own politics. The ghost of Boris Johnson's great-grandfather, Ali Kemal Bey, a liberal martyr in Turkey, might smile to find his descendant popping up as a "one-man melting pot" in London.

In seeking to emphasise the "contribution" made by immigrants, Damian Green might also have indicated that it is by no means new. But antique migrants also refused to follow obedient career paths. Friedrich Engels was the Hegel-reading son of a rich German textile family when he came to Manchester in 1842; two years later he wrote The Condition of the Working Class in England and the rest, as they say, is politics.

Background is not everything – we are not prisoners of our past. But the prominence of these migrant stories is not a coincidence either. Contrary to the notion that migrants are needy scroungers, they reveal the extent to which migration is an upmarket manoeuvre. It takes ambition, resources and drive to leap continents. Migration suits not the hapless but the exact opposite: risk-takers and careerists. In seeking to create a league table for applicants, Green risks killing the golden goose.

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