I think that Carrol and Brown are both right, but that both miss a key element of immigration (more so internal than foreign). When I (and some students) studied this issue 20 years ago, I found that the economists' statistics showed a push-pull process. Out-migration depends on rural crises of one kind or another plus a growth in the demand for laborpower in the cities and industrial districts. However, the economists as usual ignored or misread sociological factors. One study for example showed that people move to where their friends and family previously moved to. This was interpreted in terms of a market information model, i.e., spread on info on job chances and wages. In fact, each ethnic immigrant group (that we were able to study) sets up its own "welfare system" in their new "host country," which has an independent effect on migration. This created a bigger labor pool than a strict push-pull economics theory would predict. Thanks to racism, newcomers weren't welcomed into the already established welfare system offered, hence had to create their own welfare networks. The effect was to increase the reserve army of labor; and also to create closer ethnic/racial bonds between members of particular ethnic/racial groups.
In Europe in the 1970s, the economists' explanation for the return of tens of thousands of So. European workers from Germany and France and the Low Countries to their place of origin was ... the recession and lower demand for labor. Which is clearly true. However, there is another side. Once in Barcelona I spoke before a group of Spanish workers and unionists who said they returned to Spain when they did because of Franco's (then recent death). They said they wanted to help strengthen the already existing but still more or less underground independent labor movement. This kind of thing is not uncommon re: movements of workers between industries in the U.S. The economist (me in this case) showed that when unemployment insurance benefits increased relative to average money wages in particular industries, workers moved from highly seasonal and nonseasonal industries to take advantage of higher benefits (in highly seasonal industries, you couldn't work long enough to qualify for benefits; in nonseasonal industries, you weren't qualified because you were never unemployed). Then I met a fur cutter, a trotskyist, who explained that he (and many others, according to him) moved to moderately seasonal industries where the mix of wages and benefits was highest, because he wanted plenty of (paid) time to agitate and organize for the union. Unemployment insurance thus produced solidarity, a critique of the methodological individualism that one used to obtain the original results of the study. Jim O'Connor