When listening to spoken narratives, we must integrate information over multiple, concurrent timescales, building up from words to sentences to paragraphs 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. We hypothesized that the structure of this cortical processing hierarchy should result in an observable sequence of response lags between networks comprising the hierarchy during narrative comprehension. This study uses functional MRI to estimate the response lags between functional networks during narrative comprehension. We use intersubject cross-correlation analysis to capture network connectivity driven by the shared stimulus. We found a fixed temporal sequence of response lags—on the scale of several seconds—starting in early auditory areas, followed by language areas, the attention network, and lastly the default mode network. This gradient is consistent across eight distinct stories but absent in data acquired during rest or using a scrambled story stimulus, supporting our hypothesis that narrative construction gives rise to internetwork lags. Finally, we build a simple computational model for the neural dynamics underlying the construction of nested narrative features. Our simulations illustrate how the gradual accumulation of information within the boundaries of nested linguistic events, accompanied by increased activity at each level of the processing hierarchy, can give rise to the observed lag gradient.