Cloud snapshot lineage mobility between virtualization software running on different storage systems

An apparatus comprises a processing device configured to identify a cloud snapshot lineage that is being managed by first virtualization software running on a first storage system, the cloud snapshot lineage comprising one or more snapshots of at least one storage volume, the cloud snapshot lineage being stored on cloud storage of at least one cloud external to the first storage system. The processing device is also configured to pause management of the cloud snapshot lineage by the first virtualization software running on the first storage system, to obtain, from the first virtualization software running on the first storage system, configuration data for the cloud snapshot lineage, to provide, to second virtualization software running on a second storage system, the configuration data for the cloud snapshot lineage, and to resume management of the cloud snapshot lineage by the second virtualization software running on the second storage system.