Storage block balancing using volume part migration

Techniques are provided for implementing storage block balancing using volume part migration. One method comprises obtaining a capacity utilization of storage blocks each storing a portion of one or more storage volumes; and when the capacity utilization of a given storage block is above an upper target threshold, migrating a volume part of the given storage block to a different storage block having a capacity utilization below a lower target threshold, wherein the different storage block is within a same storage pool as the given storage block, and wherein the volume part comprises a subset of a storage volume. A volume part from a first storage block may also (or alternatively) be migrated to a second storage block when: (i) a number of blocks exceed a first threshold; (ii) a plurality of volume parts of the first storage block fit in one or more additional storage blocks in the same storage pool that are below the lower target threshold; and/or (iii) a storage block having a lowest utilization in the storage pool is smaller than a second threshold.