[frozenport]

Do you think these companies have contracts with all the banks, everywhere in the world?

The back-end is made from ACH/SWIFT which are indeed from traditional financial services. Access to these key services is restricted for liability reasons, although you can indeed make a wiretransfer from most bank accounts online. I can do 4 of these a month without any charges.

[masklinn]

In much of Europe, there's more or less no limit. In some countries like Belgium, direct bank transfer is so common[0] your IBAN is printed on your credit card. Others are significantly less enlightened in that regard.

[0] as in, it's a perfectly normal way to reimburse friends or colleagues who've paid stuff for you