Granular mixing is a vital operation in food, chemical, and pharmaceutical industries. Although the tumbling blender is by far the most common device used to mix grains, surprisingly little is known about mixing or segregation in these devices. In this paper, we report the first fully three-dimensional (3D) particle dynamics simulations of granular dynamics in two standard industrial tumbling blender geometries: the double-cone and the V-blender. Simulations for both monodisperse and bidisperse (segregating) grain sizes are performed and compared with experiment. Mixing and transport patterns are studied, and we find in both tumblers that the dominant mixing mechanism, azimuthal convection, contends against the dominant bottleneck, axial dispersion. The dynamics of blending, on the other hand, differs dramatically between the two tumblers: now in the double-cone is nearly continuous and steady, while flow in the V-blender is intermittent and consists of two very distinct processes.