How does connectivity impact network dynamics? We address this question by linking network characteristics on two scales. On the global scale, we consider the coherence of overall network dynamics. We show that such global coherence in activity can often be predicted from the local structure of the network. To characterize local network structure, we use "motif cumulants," a measure of the deviation of pathway counts from those expected in a minimal probabilistic network model. We extend previous results in three ways. First, we give acombinatorial formulation of motif cumulants that relates to the allied concept in probability theory. Second, we show that the link between global network dynamics and local network architecture is strongly affected by heterogeneity in network connectivity. However, we introduce a network-partitioning method that recovers a tight relationship between architecture and dynamics. Third, for a particular set of models, we generalize the underlying theory to treat dynamical coherence at arbitrary orders (i.e., triplet correlations and beyond). We show that at any order, only a highly restricted set of motifs impacts dynamical correlations.