Methods, apparatus and computer program products implement embodiments of the present invention that include configuring, by a source storage system, a logical volume to comprise source regions of the source storage system that have a mapping between a host computer and the source storage system, the source regions storing source data. The source data is copied from the source regions to target data in target regions of a target storage system, the target regions being in a one-to-one correspondence with the source regions, and the volume is reconfigured so that the target regions have the mapping between the host computer and the source storage system. In some embodiments, the mapping between the host computer and the source storage system comprises a first mapping, and upon reconfiguring the volume, a second mapping can be added between the host computer and the target storage system, and the first mapping can be deleted.