A method within a data processing system by which a processor executes an asynchronous memory move (AMM) store (ST) instruction to complete a corresponding AMM operation in parallel with an ongoing (not yet completed), previously issued barrier operation. The processor receives the AMM ST instruction after executing the barrier operation (or SYNC instruction) and before the completion of the barrier operation or SYNC on the system fabric. The processor continues executing the AMM ST instruction, which performs a move in virtual address space and then triggers the generation of the AMM operation. The AMM operation proceeds while the barrier operation continues, independent of the processor. The processor stops further execution of all other memory access requests, excluding AMM ST instructions that are received after the barrier operation, but before completion of the barrier operation.