In meandering rivers cut into bedrock, erosion across a channel cross-section can be strongly asymmetric. At a meander apex, deep undercutting of the outer bank can result in the formation of a hanging cliff (which may drive hillslope failure), whereas the inner bank adjoins a slip-off slope that connects to the hillslope itself. Here we propose a physically-based model for predicting channel planform migration and incision, point bar and slip-off slope formation, bedrock abrasion, the spatial distribution of alluvial cover, and adaptation of channel width in a mixed bedrock-alluvial channel. We simplify the analysis by considering a numerical model of steady, uniform bend flow satisfying cyclic boundary conditions. Thus in our analysis, ‘sediment supply’, i.e. the total volume of alluvium in the system, is conserved. In our numerical simulations, the migration rate of the outer bank is a specified parameter. Our simulations demonstrate the existence of an approximate state of dynamic equilibrium corresponding to a near-solution of permanent form in which a bend of constant curvature, width, cross-sectional shape and alluvial cover distribution migrates diagonally downward at constant speed, leaving a bedrock equivalent of a point bar on the inside of the bend. Channel width is set internally by the processes of migration and incision. We find that equilibrium width increases with increasing sediment supply, but is insensitive to outer bank migration rate. The slope of the bedrock point bar varies inversely with both outer bank migration rate and sediment supply. Although the migration rate of the outer bank is externally imposed here, we discuss a model modification that would allow lateral side-wall abrasion to be treated in a manner similar to the process of bedrock incision. Copyright © 2016 John Wiley & Sons, Ltd.