A storage system in one embodiment comprises a plurality of storage devices and a storage controller. The storage system is configured to participate as a target storage system in a replication process with a source storage system. In conjunction with the replication process, the target storage system is configured to receive from the source storage system replication data for at least one storage volume subject to replication from the source storage system to the target storage system, to generate a first snapshot for the storage volume, to monitor additional replication data received from the source storage system for the storage volume after generation of the first snapshot, and responsive to the monitored additional replication data satisfying one or more specified conditions, to perform at least one of the following operations: (i) generating a subsequent snapshot for the storage volume; and (ii) marking the first snapshot with a priority indicator.