>This should be pretty obvious because any money sent overseas stops being multiplied, even if you end up spending the same amount of money total.
Even given the above, I have no idea why you would assume this to be true. If someone in a foreign country is paid in US Dollars, the most likely thing to do with it is buy US goods or assets. (In the case of China, that is most likely US securities, but it's still not lost to the US economy).
Again, the multiplier effect is only applicable when dealing with an external input to a system. Otherwise, you are arguing that simply raising prices creates economic growth which is clearly not true.