When listening to spoken narratives, we must integrate information over multiple, concurrent timescales, building up from words to phrases to sentences to a coherent narrative. Recent evidence suggests that the brain relies on a chain of hierarchically organized areas with increasing temporal receptive windows to process naturalistic narratives. In this study, we use inter-subject functional connectivity to reveal a stimulus-driven information flow along the cortical hierarchy. Using cross-correlation analysis to estimate the time lags between six functional networks, we found a fixed temporal sequence of information flow, starting in early auditory areas, followed language areas, the attention network, and lastly the default mode network. This gradient is consistent across eight distinct stories but absent in resting-state and scrambled story data, indicating that the lag gradient reflects the construction of narrative features. Finally, we simulate a variety of narrative integration models and demonstrate that nested narrative structure along with the gradual accumulation of information within the boundaries of linguistic events at each level of the processing hierarchy is sufficient to reproduce the lag gradient. Taken together, this study provides a computational framework for how information flows along the cortical hierarchy during narrative comprehension.