Banks don’t run background checks when someone opens an account (so criminals have bank accounts too), however banks cooperate if requested and would signal transactions above a certain limit (I suppose) depending on the country’s laws. So many people are careful on how much money they hold in their bank accounts in comparison with what they legally earn to avoid getting in trouble. 
Anyways, this would apply on anyone running a money transfer business that is also responsible for safeguarding the money of their clients because it must be refunded to the clients if lost by the business or bank.
Of course the government have to comply with the law and lawyers usually find ways to get away with most charges if clever enough and get paid generously. Also, any mistake in the process is against the law enforcement and can cause dropping the charges in some cases.