The eastward growth of the Tibetan Plateau has been attributed to the flow of the plateau's weak lower crust into its foreland over a distance of 1500 km in 15 m.y. This channel-flow extrusion requires a very low-viscosity deep crust prior to thickening. Here, we show through triaxial thin sheet models that Tibet's uplift rate and plateau elevation demand a prethickening Moho temperature of 500-600 °C. Such temperatures are incompatible with the viscosity necessary for channel-flow extrusion >1000 km. Using two-dimensional coupled thermomechanical numerical experiments and prethickening temperatures compatible with Tibet's uplift history, we show that mass redistribution processes are dynamically coupled and that coupling is sensitive to rheology, channel buoyancy, and boundary conditions. Channel-flow extrusion velocities are limited to less than 1 cm yr-1 by cooling in the foreland and by any upward deviation of the weak channel by extension in the plateau or by erosion at the plateau margin.