Remember that many (most?) recipients of international remittances don't have bank accounts or access to payment cards. Hence why Western Union et al have to maintain extensive agent networks.
The thing to remember is that Visa authorisations are done in near real-time but settlement between issuers, Visa, acquirers and on to merchants is done over the normal banking system, as far as I know.
I don't know enough about Visa's design to comment authoritatively - but my working assumption is payments from issuing to acquiring banks in-country are done net and through visa - i.e. one net payment per issuer into visa and one net payment out to acquirer. e.g. see the last page of this doc: https://usa.visa.com/download/merchants/visa-core-principles...
What isn't obvious to me is what happens in international scenarios - you ask a great question.
As for visa, I wrote about it recently here: http://gendal.wordpress.com/2014/07/05/why-the-payment-card-...
The thing to remember is that Visa authorisations are done in near real-time but settlement between issuers, Visa, acquirers and on to merchants is done over the normal banking system, as far as I know.