The cytoplasm of a living cell is a dynamic environment through which intracellular components must move and mix. In motile, rapidly deforming cells such as human neutrophils, bulk cytoplasmic flow couples cell deformation to the transport and dispersion of cytoplasmic particles. Using particle-tracking measurements in live neutrophil-like cells, we demonstrate that fluid flow associated with the cell deformation contributes to the motion of small acidic organelles, dominating over diffusion on timescales above a few seconds. We then use a general physical model of particle dispersion in a deforming fluid domain to show that transport of organelle-sized particles between the cell periphery and the bulk can be enhanced by dynamic deformation comparable to that observed in neutrophils. Our results implicate an important mechanism contributing to organelle transport in these motile cells: cytoplasmic flow driven by cell shape deformation