Hmm, I figured that they would have essentially a "float" in most major denominations and use internal account methods to track the money and authorize it from each account. That way when you send money say from the UK to the US, the money doesn't actually get transferred from GBP to USD (and have the associated fee's), it would simply be a deposit in GBP and a withdrawl in USD, later to be cancelled out (on average) by someone sending money from the US to the UK.
You're re-inventing the Islamic Halawa transfer system:
https://en.wikipedia.org/wiki/Hawala
Money in the West works differently. We can be open about charging interest, for example.
Western Union is probably already using some kind of "Halawa" internally.