A few years back the Oracle ZFS storage had a feature called Shadow Copy. It was a pretty good method to move data from any NFS storage into the ZFS storage. Basically, you would create a new volume on the ZFS storage and point it at the source. In the background it would copy data to the ZFS storage and proxy any requests through to the old storage for data that was not yet migrated to ZFS. This was how we migrated a number of Oracle databases with minimal downtime. This only worked for NFS. CIFS or block storage still had to be copied at the client level.
--Brian