The knowledge of structural controls of river networks (RNs) on transport dynamics is important for modeling and predicting environmental fluxes. To investigate impacts of RN's topology on transport processes, we introduce a systematic framework based on the concept of dynamic clusters, where the connectivity of subcatchments is assessed according to two complementary criteria: minimum‐ and maximum‐flow connectivity. Our analysis from simple synthetic RNs and several natural river basins across the United States reveals the key topological features underlying the efficiency of flux transport and aggregation. Namely, the timing of basin‐scale connectivity at low‐flow conditions is controlled by the abundance of topologically asymmetric junctions (side‐branching), which at the same time, result in a slow‐down of the flux convergence at the outlet (maximum‐flow). Our results, when compared with observed topological trends in RNs as a function of climate, indicate that humid basins exhibit topologies which are "naturally engineered" to slow‐down fluxes. Plain Language Summary: In this study, we develop a systematic framework that characterizes the evolution of flux movement on a river network and quantifies the control exerted by the channel structure and connectivity on the environmental transport processes. Using both synthetically generated river networks as well as several natural river basins with minimum human impact across the United States, we explore the relative role of branching channels versus side‐tributaries. Our results show that the abundance of side‐tributaries have opposite implications on the timing at which the basin achieves full connectivity: faster convergence at minimum‐flow conditions, while slower aggregation of fluxes at the outlet (peak flow) of the river network. Our study indicates that the connectivity of river networks, and therefore their efficiency in aggregating fluxes, differ in humid and arid environments. Key Points: A framework to characterize the dynamic connectivity emerging from the transport of fluxes on river networks is introducedThe abundance of side‐branching junctions (accelerates) decelerates the timing to full connectivity under (low) peak flow conditionsHumid basins exhibit topologies which are prone to slow‐down the convergence of fluxes [ABSTRACT FROM AUTHOR]