We construct a spatiotemporal frame for the study of spatial economic and ecological patterns generated by transboundary pollution. Space is continuous and polluting emissions originate in the intensity of use of the production input. Pollution flows across locations following a diffusion process. The objective functional of the economy is to set the optimal production policy over time and space to maximize welfare from consumption, taking into account a negative local pollution externality and the diffusive nature of pollution. Our framework allows for space and time dependent preferences and productivity, and does not restrict diffusion speed to be space-independent. Accordingly, we develop a methodology to investigate the environmental and economic implications of spatiotemporal heterogeneity. We propose a method for an analytical characterization of the optimal paths. An application to technological spillovers is proposed for illustration. We focus on the determination of the optimal short-term spatiotemporal dynamics induced by the resulting non-autonomous problems.