Australia has a lot of foreign businesses and it has a lot of immigrants. Both earn Australian dollars and huge amounts would be sent back their country of origin.
His does Australia balance its books on something like this? How do the economics of it work? Would it lower Australian inflation but shortening the money supply, and raise inflation of the destination country as it prints more money to exchange the Australian dollar?
That 500AUD doesn’t just sit in an account and magically contribute to anything.
Currency exchange doesn’t actually happen in a vacuum. The only reason your Bangladeshi example is able to send 500AUD to his family, is that someone who has Bangladeshi Taka wants 500AUD to buy goods or services from somewhere that accepts AUD. And there’s a very short list of countries that spend AUD.
So that 500 doesn’t disappear to never return. That 500 is sold to someone who wants to use it to purchase australian exports.