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Every year, Mexicans working in the U.S. send at least $20 billion back to Mexico in the form of remittances, placing a huge drain on our economy.
But I don't see what negative effects it could have on the US economy. Won't the money find their way back to the US, since that is the only place where you are able to trade them for goods and services?
Would the impact be different for a smaller country with a currency not as global as the US dollar?
The argument for the "draining the economy" is that money is not recirculated in the economy, and cannot be spent in consumption, which generates growth.
It would be a similar argument, however, for several other situations, like the firms implementing automation: the shareholders will capture the wealth from wages saved. The portion of this money that is saved in personal fortunes is also not being recirculated in the economy, slowing down growth.
As the user london pointed out in the comments - the size of the US economy makes the remittances insignificant. I assume this is the case for most large countries.
If you want to explore a case of a small country with large remittances, I believe Qatar is one of the big outliers out there, where the migrant population is 88% of the total population of the country, sending $11B in remittances in 2016.
Whatever money earned in a country that is spent outside that country is taking away economic activity from that country, GDP can no longer be increased with that money.
Sending money to a country with a different currency usually implies that currency is bought. That will increase the value of the bought currency and decrease the value of the currency sold. A lower US dollar will make it easier for US exporters to sell their products abroad, which will lead to foreign importers selling their foreign currency in return for US dollars. That is how the money can come back at some point.
It can also come back if foreigners invest in the US. If the US has a lot of money from remittances flowing out and doesn't export as much as it imports, which has been the case for a long time, foreigners holding US stocks and bonds will be the result. You could argue this could lead to a dangerous dependency at some point, e.g., if you look at how much the US owes to the Chinese. Mexico in comparison is too small an economy to be such a threat and I agree with the comment above that losing a significant number of Mexican workers would be much more problematic to the US economy.
You should also take into account that the positive development effects of remittances in the receiving country should that country to become a more attractive partner for trade with the sending country, especially given the strong relationship that is built by the migrant community.
