Experimentally determined shoreline migration rates show high-frequency autogenic variability superimposed on low-frequency allogenic shoreline responses induced by eustatic base level change. This variability persists even when the shoreline migration is averaged laterally, indicating time variation in total sediment discharge reaching the shoreline. The magnitude of autogenic variability in shoreline migration rate changes by roughly a factor of 3 depending on the shoreline migration direction: It is strongest during transgression, when the shoreline is on average migrating against the mean sediment flow, and weakest during regression, when the shoreline is migrating with the mean sediment flow. We propose that this time variation is due to overall sediment storage and release in the fluvial system. We use a one-dimensional geometric model to model the autogenic signals observed in the experiment. The model uses small periodic changes in the fluvial slope to represent the effects of storage and release of supplied sediment due to intrinsic variation in the transport efficiency of the fluvial system. The periodic pulses of sediment discharge to the shoreline required in the model to explain the variations in shoreline migration can exceed by an order of magnitude the mean allogenic sediment discharge. The slope variability required in the model to explain the autogenic shoreline signals is 1--4% of the mean slope. This is well within the range of observed variability in depositional fluvial slopes. We propose that at least part of this observed variability is real; that is, the long-profile slope of a depositional river system may have an intrinsic "elasticity" of a few percent of its mean value, even under steady forcing. Autogenic slope variation of this magnitude could readily produce parasequence-scale deposits in the stratigraphic record.