A 10-minute walk from the Bank of England, on the eastern edge of the City of London, lies a gateway to a new shadow world of money.

Here on Dukes Place is the office of Moorwand Ltd., one of a fast-growing breed of upstarts that bill themselves as alternatives to old-fashioned banks when moving money around the world. Each day in the U.K. alone, an estimated 1.4 billion pounds ($1.9 billion) courses through loosely regulated digital payments businesses like Moorwand. Though only a small fraction of Britain’s financial flows, it’s a system critics warn is opening a door for dirty money.