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Gravel‐bed rivers that incise into bedrock are common worldwide. These systems have many similarities with other alluvial channels: they transport large amounts of sediment and adjust their forms in response to discharge and sediment supply. At the same time, the occurrence of bedrock incision implies behaviour that falls on a spectrum between fully detachment‐limited ‘bedrock channels’ and fully transport‐limited ‘alluvial channels’. Here, we present a mathematical model of river profile evolution that integrates bedrock erosion, gravel transport and the formation of channels whose hydraulic geometry is consistent with that of near‐threshold alluvial channels. We combine theory for five interrelated processes: bedload sediment transport in equilibrium gravel‐bed channels, channel width adjustment to flow and sediment characteristics, abrasion of bedrock by mobile sediment, plucking of bedrock and progressive loss of gravel‐sized sediment due to grain attrition. This model contributes to a growing class of models that seek to capture the dynamics of both bedrock incision and alluvial sediment transport. We demonstrate the model's ability to reproduce expected fluvial features such as inverse power law scaling between slope and area, and width and depth consistent with near‐threshold channel theory, and we discuss the role of sediment characteristics in influencing the mode of channel behaviour, erosional mechanism, channel steepness and profile concavity.