A storage system includes a three-dimensional memory array that has multiple layers of non-volatile storage elements grouped into blocks. Each block includes a subset of first selection circuits for selectively coupling a subset of array lines (e.g. bit lines) of a first type to respective local data lines. Each block includes a subset of second selection circuits for selectively coupling a subset of the respective local data lines to global data lines that are connected to control circuitry. To increase the performance of memory operations, the second selections circuits can change their selections independently of each other. For example, a memory operation is performed concurrently on a first non-volatile storage element of each group of a plurality of groups of non-volatile storage elements. Completion of the memory operation for the first non-volatile storage element of each group is independently detected. A memory operation on a second non-volatile storage element of each group is independently commenced for each group upon independently detecting completion of the memory operation for the first non-volatile storage element of the respective group.