1. A method of operating a data storage system, comprising: receiving a host I/O request from a host-side interface of the data storage system, the host I/O request specifying a range of logical block addresses (LBAs) of a mapped logical unit (MLU) of storage presented to an external host computer by the host-side interface, the MLU being mapped through a device-side interface to a plurality of underlying logical units of storage; obtaining, in response to receiving the host I/O request, mapping information for a plurality of extents of the underlying logical units of storage, the extents being mapped to respective sub-ranges of the range of LBAs, the host-side interface being configured to transfer the extents during a subsequent transfer phase; upon obtaining the mapping information, determining whether there are a sufficient number of free sub-I/O request tracking structures to track completion of a plurality of respective sub-I/O requests to be used for the extents of the underlying logical units of storage, and only upon there being such a sufficient number of free sub-I/O request tracking structures, then concurrently issuing to the device-side interface, using the mapping information, the plurality of sub-I/O requests for the extents of the underlying logical units of storage, the device-side interface providing respective transfer initiation responses for the sub-I/O requests to initiate transfer of the respective extents; receiving the transfer initiation responses from the device-side interface and forwarding the transfer initiation responses to the host-side interface to cause the respective extents to be transferred by the host-side interface in the transfer phase; and as the transfer initiation responses are forwarded to the host-side interface, freeing the respective sub-I/O request tracking structures for use in processing subsequent host I/O requests.