The present analysis derives a stability criterion for long-term equilibrium channel heads. The concept of finite perturbation analysis is presented, during which the surface is subjected to perturbations of a finite amplitude and resulting changes in flow path structure and slope are computed. Based on these quantities the analysis predicts whether the perturbed location is going to erode, be filled in or remain steady. The channel head is defined geometrically as the focus point of converging flow lines at the bottom of hollows. It is demonstrated that stability at the channel head grows out of the competition between the rate of flow path convergence and the degree of profile concavity. Analytical functions are derived to compute channel head-contributing area and -slope, flow path convergence and profile concavity as a function of perturbation depth, distance from the crest and the initial slope. In a numerical model these quantities point to the long-term equilibrium channel head position, which is shown to depend also on the width to length ratio of hollows. It is also demonstrated that the equilibrium channel head position is sensitive to the base-level lowering/non-dimensional slope length ratio and to the slope of the initial topography. Morphometrical measurements both in the field and on simulated topographies were used to test the theoretical predictions.