The evolution of rivers in eroding landscapes plays a key role in determining landscape relief and modulating climate‐tectonic interactions. A common approach to quantifying river system evolution uses a one‐dimensional, detachment‐limited stream power equation. One potential drawback of this model is that it does not incorporate the effects of changes in channel width or the role of sediment transport dynamics. Here I present a new method for modeling the influence of channel width on river dynamics to explore how variable width and sediment transport impact river profile evolution. With this approach, vertical river erosion can operate based on any number of river erosion models, such as a simple shear stress model (e.g., detachment limited), sediment cover‐shear stress hybrid models, or mechanistic saltation‐abrasion models. I explore the sensitivity of these three models to increases in rock‐uplift rate (i.e., 2, 3, 5, 10, and 20× increase). Generally, the results show that incorporating channel width adjustment or sediment transport dynamics lowers the sensitivity of a river profile to rock‐uplift rate. For the sediment transport‐dependent models, the degree of sensitivity depends on whether the system is limited by bedrock exposure or erosion potential (i.e., detachment potential). The approach produces transient responses that reveal distinct patterns of width and slope, which may provide valuable insight into the limiting physical mechanisms of bedrock erosion in a region. The implications of the work are broad and include the potential to distinguish underlying erosion controls from field observations of width and slope as well as understanding climate‐tectonic interactions.