No. If their transactions add up to a net outflow, banks use the money that is already in their BoE accounts to make transactions in favour of the BoE accounts

No. If their transactions add up to a net outflow, banks use the money that is already in their BoE accounts to make transactions in favour of the BoE accounts of other banks or the FPS (those where the net outflow went to). Since this does not change the total amount of money at BoE accounts, it does not affect the parent's question about where that money originally came from.