LISA RAKOCZY arrived in north London in the mid-2000s from Krakow to study English, hoping to support herself by working as a cleaner. All went well—until the crash and ensuing economic crisis hit her previously prosperous clients. Competition for jobs became fiercer as middle-class families started to spend less on household help. For a while, she eked out enough to pay for her language course by cleaning lavatories in railway stations. Later, her sister, a teacher in Poland, helped her to make ends meet. But by the end of last year, Ms Rakoczy had tired of the trials of an immigrant's life and headed home.

A typical case, you might think. Indeed, in the wake of the crash, many immigrants have gone back. Just as predictably, many would-be emigrants have stayed at home, either because moving abroad no longer seems worth the effort or because immigration rules in many countries have become more restrictive.

Yet at the airport, Ms Rakoczy may have crossed a young Briton bound for Shanghai, a Chinese computer programmer moving to Canada or a Portuguese worker on his way to oil-rich Norway. The recession has not stopped all migration, but rather led to new patterns and different destinations. Never before—or at least not in recent history—has the map of global migration been at the same time so varied and so changeable.

Liberalisation of travel after the end of the cold war, the West's economic boom of the 2000s and rapid growth of emerging markets—all of these contributed to a new surge of migration until the onset of the economic crisis. But as unemployment has risen, governments have grown more sensitive to arguments that immigration can be a drain on public services and damage the job prospects of the native population.

Take Britain, where the government has introduced a “migration cap” for workers coming from outside the European Union. David Cameron, the prime minister, has said that the aim is to cut unskilled immigration. A “points system” is meant to identify immigrants who will be most useful to the economy. In addition, the student visa regime is to be tightened. (Only the churlish would point out that both he and Nick Clegg, his coalition partner, were themselves beneficiaries of time spent abroad before studying.) Britain may be going further than other countries, but it is by no means alone. Young east European democracies such as Poland are piloting schemes to restrict the numbers of incomers. The Danish government has recently reintroduced limited border controls in defiance of the European Union's Schengen agreement, which permits passport-free travel between 23 of the 27 EU countries. The government claims this measure was needed to combat crime and smuggling, though many suspect that it is really about pandering to the anti-immigration Danish People's Party, on which it relies for support. Spain, Denmark and Japan are among those countries giving a new meaning to the phrase “pay as you go”: they compensate workers who agree to return to their country of origin. The effect of such measures seems limited. Up to April 2010, only 11,400 immigrants in Spain had taken up the offer. This summer, the government in Madrid reintroduced controls on Romanian immigrants. And France has sought to deport Romanian gypsies. Other governments, more squeamish about targeting distinct ethnic minorities or social groups, have tried to revive the “buffer theory”—that migrants return home when the economy contracts, freeing jobs for the native population and providing what some call a “conjunctural shock absorber”. To boost this effect, Australia has cut the number of skilled worker visas that it offers. Malaysia more or less stopped issuing work permits in 2009.

Yet the “buffer theory” seems to be based on weak empirical foundations. A recent study by Policy Network, a think-tank, concluded that trends in unemployment and income have only a modest impact on net immigration. Using data from the past three recessions in Britain, they reckon that immigration falls when unemployment rises—but only for a limited period, after which it picks up again, often before there is any clear improvement in the economy. Similarly, the number of returning immigrants goes up only temporarily, when those who always intended to return do so.

Still, the general picture now is of less migration. The OECD, an international think-tank, says that migration into its member countries dropped by 7% in 2009. Recent national data suggest that migration will have fallen further in 2010. Dig deeper, however, and the position gets more complicated. Ireland was particularly attractive during its banking and building boom. But in 2009 its net migration was negative. Ireland now has its highest outward migration since 1989.

Australia is also a magnet for young Chinese, the largest group of immigrants in 2010. Sweden, too, is proving increasingly popular. Meanwhile, Greece, plagued by an unattractive combination of fiscal problems, austerity and unemployment, has seen a new wave of emigration, notably to America and Canada. As for Spain, modestly prosperous Britons and Germans no longer see the Costa del Sunshine as their ideal retirement resort.

“Circular” migration, in which people come and go between destinations, is on the rise, as is “on-migration”, where a migrant moves first from China to Canada, say, and then on to America. OECD researchers reckon that at least 19% of migrants who arrived in America at the turn of the millennium had left for other destinations five years later. On-migration is also common among migrants from Africa and Asia. Europeans, for their part, tend to live abroad for only a limited time.

“The notion that migration is a one-way movement of permanent settlement is outdated. Most of it is temporary—and it's time the debate about immigration recognised this reality,” argues Philippe Legrain, an analyst of immigration and the author of “Aftershock”, a recent book analysing economic changes in the wake of the financial crash.

Some countries are starting to recognise this. While sounding tough, politicians are tweaking earlier decisions intended to restrict immigration. Germany, which has long insisted that it is not an “immigrant country”, has nonetheless quietly increased the number of skilled migrants it allows in. Sweden has liberalised its work-permit system.

An even more intriguing development is that America and China have begun to reverse roles. America has always been the greatest immigration magnet of all. But stricter security measures to thwart terrorist attacks and the severity of its recession have changed this. Both legal and illegal immigration have tumbled, with the greatest decrease among Mexican illegal immigrants (only partly due to stricter border controls, as potential immigrants can now also find better jobs at home).

Go East

At the more educated end of the market, foreign-born college graduates are increasingly likely to leave America after gaining skills and qualifications. Some of this may be cyclical: they may return when the economy picks up. Yet in the current climate, travellers have a good chance of meeting an Asian-born graduate from an American college who is moving somewhere else to work.

Even as America's allure is fading, China is becoming a destination of choice for many young workers. According to Chinese statistics, last year Shanghai had 143,000 foreigners with residents' visas. That does not count the many thousands of Westerners believed to be there on tourism visas, or the illegals from elsewhere in Asia. South Koreans (121,000) top the list of expatriates resident in China, followed by Americans (71,000) and Japanese (66,000). Teaching English is the commonest job for Westerners, but there are also many, usually young, entrepreneurs opening shops, bars and restaurants.

The boom in China and the country's activities in Africa have also encouraged more Africans to consider seeking their fortune in the Middle Kingdom. Some 100,000 are settled in Guangzhou. But African immigrants are not the only ones who wind up in this bustling city in southern China. One recent academic study identified five different residential zones of immigrant populations. African traders and Koreans, for instance, live in crowded districts. French and Indian workers congregate in high-rise buildings. Successful immigrants from the Middle East and west Africa cluster in a large white-collar estate with private gardens.

The world may be witnessing the beginnings of a big trek East. Students have started to move to South Korea and Japan. “Many international students remain in their host country,” says Madeleine Sumption, a researcher at the Migration Policy Institute in Washington, DC. She expects this trend to fuel growth in migration between Asian countries.

Far from disappearing in the wake of the crash, Ms Sumption says, migration is still “a sensible long-term investment for many people.” Although hard times may change migrants' destinations, they do not sap the will to move in search of a better life. This is good news: migrants did not contribute to the economic crisis, and they may yet help to overcome it.