Benefits, costs, and requirements accompany the transition from motile totipotent unicellular organisms to multicellular organisms having cells specialized into reproductive (germ) and vegetative (sterile soma) functions such as motility. In flagellated colonial organisms such as the volvocalean green algae, organized beating by the somatic cells' flagella yields propulsion important in phototaxis and chemotaxis. It has not been generally appreciated that for the larger colonies flagellar stirring of boundary layers and remote transport are fundamental for maintaining a sufficient rate of metabolite turnover, one not attainable by diffusive transport alone. Here, we describe experiments that quantify the role of advective dynamics in enhancing productivity in germ soma-differentiated colonies. First, experiments with suspended deflagellated colonies of Volvox carteri show that forced advection improves productivity. Second, particle imaging velocimetry of fluid motion around colonies immobilized by micropipette aspiration reveals flow fields with very large characteristic velocities U extending to length scales exceeding the colony radius R. For a typical metabolite diffusion constant D, the associated Peclet number Pe = 2UR/D >> 1, indicative of the dominance of advection over diffusion, with striking augmentation at the cell division stage. Near the colony surface, flows generated by flagella can be chaotic, exhibiting mixing due to stretching and folding. These results imply that hydrodynamic transport external to colonies provides a crucial boundary condition, a source for supplying internal diffusional dynamics.