Spatial networks are known to be informative about the spatiotemporal transmission dynamics of COVID-19. Using district-level panel data from Germany that cover the first 22 weeks of 2020, we show that mobility, commuter and social networks all predict the spatiotemporal propagation of the epidemic. The main innovation of our approach is that it incorporates the whole network and updated information on case numbers across districts over time. We find that when disease incidence increases in network neighbouring regions, case numbers in the home district surge one week later. The magnitude of these network transmission effects is comparable to within-district transmission, illustrating the importance of networks as drivers of local disease dynamics. After the introduction of containment policies in mid-March, network transmission intensity drops substantially. Our analysis suggests that this reduction is primarily due to a change in quality—not quantity—of interregional movements. This implies that blanket mobility restrictions are not a prerequisite for containing the interregional spread of COVID-19.