Management of data written via a bus interface to a storage controller during consistent copying of data

Remote copy operations are performed to copy data from a primary storage controller to a secondary storage controller, wherein input/output (I/O) requests are received at the primary storage controller from a host both via a bus interface and a network interface while the remote copy operations are in progress, and wherein consistency groups are formed during the remote copy operations to copy the data consistently. Quiescing of I/O operations performed via the bus interface are performed while a current consistency group is being replaced by a next consistency group during the remote copy operations.