Product : VMware, vSphere/6.0, Enterprise Plus
Feature : Storage Migration, VM Mobility, VM Mobility and HA
Content Owner:  Roman Macek
Summary
Yes (Live Storage vMotion); including replicated VMs
Details
Storage vMotion allows to perform live migration of virtual machine disk files (e.g. across heterogeneous storage arrays) without vm downtime. Storage DRS handles initial vmdk placement and gives migration recommendations to avoid I/O and space utilization bottlenecks on the datastores in the cluster. The migration is performed using storage vMotion

With vSphere 5.5. the number of concurrent Storage vMotion operations per datastore is 8 and the number of concurrent Storage vMotion operations per host 2.
New in vSphere 5.5 is the ability to migrate VMs replicated with vSphere Replication (using Storage vMotion or Storage DRS) - note that Storage vMotion is only supported for the replicated VM, not the target replica.

Since vSphere 5.1 Storage DRS is compatible with vCloud Director.

vSphere 5 introduced support to migrate virtual machines that have snapshots/linked clones and improved the migration performance using i/o mirroring. vSphere 5 also introduced Storage DRS (enhanced, automated storage VMotion) - the ability to logically pool storage resources (datastore cluster) and migrate the actual virtual machine data to other disk resources based on performance policies.