Problems of forecasting related time series of counts arise in a diverse range of applications including consumer sales, epidemiology, ecology, and law enforcement. With a focus on multi-step, on-line forecasting, we have developed new classes of models based on the concept of decouple/recouple applied to multiple series that are individually represented via novel univariate state-space models. The latter involve dynamic generalized linear models for binary and conditionally Poisson time series and dynamic random effects for overdispersion. New multivariate models enable information sharing across series when aggregated data provide more incisive inferences on shared patterns such as trends and seasonality. We extend these models to a general framework appropriate for settings in which count data arise through a compound process. Several case studies in many-item, multi-step ahead supermarket sales forecasting demonstrate improved forecasting performance using the proposed models, with discussion of forecast accuracy metrics and the benefits of probabilistic forecast accuracy assessment.