A data storage system manages use of a pool of secondary storage by internal file systems hosting logical data storage objects accessed by clients. A choose-and-mark-slices operation scans the file system to identify a subset of the slices as evacuatable slices to be de-provisioned and returned to the pool, the subset having a size exceeding a per-iteration limit observed by a separate evacuate-slices operation. The subset is recorded in an in-memory structure. Each iteration of the evacuate-slices operation (1) examines the in-memory structure to select slices for evacuation, no more than the per-iteration limit being selected, (2) moves data from the selected slices to the remaining slices of the set, and (3) subsequently (i) updates metadata of the file system to reflect the moving of the data and (ii) de-provisions the selected slices from the file-system to return them to the pool for subsequent re-provisioning.