Data is replicated from a source to a target. For each of a plurality of replication processing threads (one thread at a time) a corresponding batch of one or more non-synchronization tasks is formed. Each batch of one or more non-synchronization tasks includes a next consecutive set of non-synchronization tasks drawn from a source of non-synchronization tasks that is common to the replication processing threads included in the plurality of replication processing threads. A replication processing thread that has associated with it an outstanding batch of one or more non-synchronization tasks that the thread has not finished communicates via an associated connection to the target as needed to perform such unfinished tasks included in the outstanding batch without regard to whether other threads included in the plurality of threads are communicating with the target via one or more other connections to the target.