In this paper, we develop a model of COVID spread that uses dynamic mobility networks, derived from US cell phone data, to capture the hourly movements of millions of people from local neighborhoods (census block groups, or CBGs) to points of interest (POIs) such as restaurants, grocery stores, or religious establishments. Using these networks, we simulate COVID spread in 10 of the largest US cities, and show that even a relatively simple epidemiological model can accurately capture the COVID case trajectory despite dramatic changes in behavior due to the virus. Furthermore, by modeling detailed information about each POI, like visitor density and visit length, we can estimate the impacts of fine-grained reopening plans: we predict that a small minority of "superspreader" POIs account for a large majority of infections, that reopening some POI categories (like full-service restaurants) poses especially large risks, and that strategies restricting maximum occupancy at each POI are more effective than uniformly reducing mobility. Our models also predict higher infection rates among disadvantaged racial and socioeconomic groups solely from differences in mobility: disadvantaged groups have not been able to reduce mobility as sharply, and the POIs they visit (even within the same category) tend to be smaller, more crowded, and therefore more dangerous.