The paper introduces new methods for inference with count data registered on a set of aggregation units. Such data are omnipresent in epidemiology because of confidentiality issues: it is much more common to know the county in which an individual resides, say, than to know their exact location in space. Inference for aggregated data has traditionally made use of models for discrete spatial variation, e.g. conditional auto-regressive models. We argue that such discrete models can be improved from both a scientific and an inferential perspective by using spatiotemporally continuous models to model the aggregated counts directly. We introduce methods for delivering (limiting) continuous inference with spatiotemporal aggregated count data in which the aggregation units might change over time and are subject to uncertainty. We illustrate our methods by using two examples: from epidemiology, spatial prediction of malaria incidence in Namibia, and, from politics, forecasting voting under the proposed changes to parliamentary boundaries in the UK. © 2018 Royal Statistical Society