Seeing the forest of deleted comments I'm not sure if this passes muster, but this is really an economics question, not a history question (plus that statement is pretty much impossible to prove, we don't have good economic data much past the latest century or two and it's generally hard to prove that something has never happened). But what I think you're friend is getting at is that economists are really quite pro-immigration, and the economic case for it is rather sound. Here's an article that details the case, while I'm sure it talks about modern events I'm hoping that's okay since I'm guessing your friend doesn't really care about whether or not there's a counterexample somewhere in history, but rather that there's very strong evidence that immigration is (economically) a net positive for both the immigrant and the country.
I'm pretty sure you and your friend were discussing the recent trend of Hispanic immigration in the United States.
The problem with historical examples is that they can't easily be applied to comparisons with current labor markets. I could write about Flemish immigrants in England, who laid the foundation for the English textile industry, or European (especially white, anglo-saxon protestant) colonizers in North America, which turned the US and Canada into economic powerhouses to the detriment of their original inhabitants. But these example have little to do with modern immigration trends. The risk that white, anglo-saxon protestants will be shipped off to reservations like Native Americans were is nonexistent.
Don't be fooled by my flair. As an Italian Economic Historian, I know a thing or two about both emigration and immigration. What trends, then, are observable in relatively modern cross-border worker flows?
Firstly, Europe is currently attracting more migrants per person than the United States. SInce the postwar era, the United States has experienced declining inflows of migrants in proportion to native population, and much lower overall levels compared to the beginning of the twentieth century and at the end of the 19th century, in great part due to strict restrictions on immigration.
In Europe, on the other hand, in the late 1940's and 1950's, postwar shortages of labor encouraged various european countries to to open their labor markets and recruit foreign labor. These countries were France, Germany, the United Kingdom, Swizerland, and Belgium. (Zimmermann K.F. ed. 2005, European Migration, what do we know? Oxford, Oxford University Press) In a 2013 study of the "Best Country to be Born In" conducted by The Economist, Swizerland and Belgium are ranked higher than the United States, whereas France and The United Kingdom are ranked slightly lower than the U.S. (Germany ranks on the exact same level). Although the long-term effects of the recruitment of foreign labor is not entirely responsible for the prosperity of these countries, it is easy to argue that prosperity of attraction of migrants goes hand in hand.
But what comes first? Does prosperity attract migrants or do migrants cause prosperity?
Common economic theory suggest that labor market migration should negatively impact wages of natives, a natural consequence of the downward-sloping demand curve. This should also reduce employment among natives. The magnitude of wage reduction as opposed to unemployment has to do with elasticity of labor supply (i.e. how difficult social and legal infrastructure makes it is to adjust worker wages).
However, the above theory has not been backed up by the evidence, mainly because immigration is not a random process. It is a rational choice that depends on two decisions: where to relocate, and if it is worth it to relocate at all. In the United States, unemployment rates among foreign-born males are lower than the native population (Borjas, G.J. 1987 Self-selection and the Earnings of Immigrants American Economic Review 77 (4) 513-53). Foreign-born females, on the other hand, tend to have slightly higher unemployment levels when compared to native-born females (and can be attributed to lower labor-market participation). Further, among lower-income groups, such as Highschool dropouts, there is no significant wage fluctuation in periods and places with a large inflow of migrants (Card D. The Impact of the Mariel Boatlift on the Miami Labor Force 1990 Industrial Labor Relations Review 43:245-57)
In addition, empirical literature has not found any statistically significant correlation between wage levels and influx of migrants, even in those border areas (such as Southern European countries like Italy) where the arrival of many transitory migrants renders the unemployment levels among migrants much higher than unemployment levels among natives (Friedberg and Hunt, 1995, The Impact of Immigrants on Host Countries Wages, Employment, and Growth, Journal of Economic Perspectives 9:23-44 + studies conducted more recently that, however, breach this sub's 20-year rule).
The empirical results can be reconciled with the economic theory by taking into consideration three things:
Immigrants self-select by migrating into high-wage regions, meaning that regions with higher wage and lower unemployment are likely to attract more migrants, creating a positive correlation between migration and wages.
Changes in internal migration patterns by native workers, especially in the United States where labor is very mobile, can create labor shortages that are filled by migrants. Further, in Europe, where where regional labor mobility is much lower, foreign migrants can compensate for the lack of mobility in residents.
Open economies can adjust to migration by changing output mix or production technology to reap the benefits of increased labor supply.
Consequentially, immigrants, as a whole tend to "Grease the Wheels" of the economic system. This is true for most historic migration patterns. Flemish immigrants to England, Huguenots in colonial Luisiana, and the massive waves of non-WASP European migrants to america in the 19th century all brought their labor, skilled or unskilled, to a place where it was in demand.
Source: T. Boeri and J. van Ours The Ecomonics of Imperfect Labor Markets 1990, Princeton, Princeton University Press).