Transport of mass within cells helps maintain homeostasis and is disrupted by disease and stress. Here, we develop quantitative phase velocimetry (QPV) as a label-free approach to make the invisible flow of mass within cells visible and quantifiable. We benchmark our approach against alternative image registration methods, a theoretical error model, and synthetic data. Our method tracks not just individual labeled particles or molecules, but the entire flow of material through the cell. This enables us to measure diffusivity within distinct cell compartments using a single approach, which we use here for direct comparison of nuclear and cytoplasmic diffusivity. As a label-free method, QPV can be used for long-term tracking to capture dynamics through the cell cycle. Finally, based on the known effective particle size, we show that QPV is an accessible method to measure intracellular viscosity.