Spatially distributed collective adaptive systems are an important class of systems that pose significant challenges to modeling due to the size and complexity of their state spaces. This problem is acute when the dynamic behavior of the system must be captured, such as to predict system performance. In this article, we present an abstraction technique that automatically derives a moment-closure approximation of the dynamic behavior of a spatially distributed collective adaptive system from a discrete representation of the entities involved. The moment-closure technique is demonstrated to give accurate estimates of dynamic behavior, although the number of ordinary differential equations generated for the second-order joint moments can grow large in some cases. For these cases, we propose a rigorous model reduction technique and demonstrate its use to substantially reduce the computational effort with only limited impact on the accuracy if the reduction threshold is set appropriately. All techniques reported in this article are implemented in a tool that is freely available for download.