I've been thinking about this a lot and I am not sure I have understood correctly. I was thinking about, for example, migrant workers who send a portion of their income back home, or wealthy individuals who hold money in offshore accounts (in a foreign currency).
Very often I see these sorts of scenarios represented as money being "lost" by the source economy and gained by the destination economy. However, if one thinks about the transactions involved, what is actually happening is that value is being exchanged between two isolated economies. Similar to (to use a tired plumbing analogy) a heat exchanger which transfers heat from one plumbing system to another whilst the medium (currency) within each system never directly interacts.
If somebody transfers their money from one economy to another via an FX trade, they purchase, for example, Dollars with their Euros and they now have some Euros which represent the value previously represented by their Dollars, whilst the Dollars they previously had are taken by whomever sold the Euros and put back into the Dollar economy they came from.
Therefore, there is no change in the quantity of money in either economy, so there are no gains or losses. Have I understood correctly?
The only potential downside I can see is that there may be a subsequent imbalance in wealth distribution in the destination economy, but that's the opposite of the way it is presented where the downside is on the source economy. Does this make sense?