We construct a dynamic general equilibrium model with occupation mobility, human capital accumulation and endogenous assignment of workers to tasks to quantitatively assess the aggregate impact of automation and other task-biased technological innovations. We extend recent quantitative general equilibrium Roy models to a setting with dynamic occupational choices and human capital accumulation. We provide a set of conditions for the problem of workers to be written in recursive form and provide a sharp characterization for the optimal mobility of individual workers and for the aggregate supply of skills across occupations. We craft our dynamic Roy model in a production setting where multiple tasks within occupations are assigned to workers or machines. We solve for the balanced-growth path and characterize the aggregate transitional dynamics ensuing task-biased technological innovations. In our quantitative analysis of the impact of task-biased innovations in the U.S. since 1980, we find that they account for an increased aggregate output in the order of 75% and for a much higher dispersion in earnings. If the U.S. economy had higher barriers to mobility, it would have experienced less job polarization but substantially higher inequality and lower output as occupation mobility has provided an "escape" for the losers from automation.