Sculpting of structure and function of three-dimensional multicellular tissues depend critically on the spatial and temporal coordination of cellular physical properties, yet the organizational principles that govern these events, and their disruption in disease, remain poorly understood. Using a multicellular mammary cancer organoid model, here we map in three dimensions the spatial and temporal evolution of positions, motions, and physical characteristics of individual cells. Compared with cells in the organoid core, cells at the organoid periphery and the invasive front are found to be systematically softer, larger and more dynamic. These mechanical changes are shown to arise from supracellular fluid flow through gap junctions, suppression of which delays transition to an invasive phenotype. Together, these findings highlight the role of spatiotemporal coordination of cellular physical properties in tissue organization and disease progression.