Data on count processes arise in a variety of applications, including longitudinal, spatial and imaging studies measuring count responses. The literature on statistical models for dependent count data is dominated by models built from hierarchical Poisson components. The Poisson assumption is not warranted in many applied contexts, and hierarchical Poisson models make restrictive assumptions about overdispersion in marginal distributions. In this article we propose a class of nonparametric Bayes count process models, constructed through rounding real-valued underlying processes. The proposed class of models accommodates situations in which separate count-valued functional data are observed for each subject under study. Theoretical results on large support and posterior consistency are established, and computational algorithms are developed based on Markov chain Monte Carlo simulation. The methods are evaluated via simulation and illustrated by application to longitudinal tumour counts and to asthma inhaler usage. © 2013 Biometrika Trust.