In 2014-2016, Guinea, Sierra Leone and Liberia in West Africa experienced the largest and longest Ebola epidemic since the discovery of the virus in 1976. During the epidemic, incidence data were collected and published at increasing resolution. To monitor the epidemic as it spread within and between districts, we develop an analysis method that exploits the full spatiotemporal resolution of the data by combining a local model for time-varying effective reproduction numbers with a gravity-type model for spatial dispersion of the infection. We test this method in simulations and apply it to the weekly incidences of confirmed and probable cases per district up to June 2015, as reported by the World Health Organization. Our results indicate that, of the newly infected cases, only a small percentage, between 4% and 10%, migrates to another district, and a minority of these migrants, between 0% and 23%, leave their country. The epidemics in the three countries are found to be similar in estimated effective reproduction numbers, and in the probability of importing infection into a district. The countries might have played different roles in cross-border transmissions, although a sensitivity analysis suggests that this could also be related to underreporting. The spatiotemporal analysis method can exploit available longitudinal incidence data at different geographical locations to monitor local epidemics, determine the extent of spatial spread, reveal the contribution of local and imported cases, and identify sources of introductions in uninfected areas. With good quality data on incidence, this data-driven method can help to effectively control emerging infections.