In spite of recent development, state-of-the-art debris-flow incision models (DFIM) remain under-constrained, limiting their integration in landscape evolution model (LEM), and thus our understanding of catchment erosion dynamics, including their altitude distribution, their response to climatic and tectonic changes and drainage network reorganization. Here, we use a novel method to extract channel heads on high-resolution topographies of 38 catchments spanning two orders of magnitude of erosion rates, and constrain the gradient and concavity of hundreds of first order-channels. We show that this is consistent with a novel, refined analytical DFIM prediction and constrain the exponents relating incision, slope gradient and debris-flow thickness. We also provide constraints on the range of possible value for debris-flow erodibility, as a function of debris-flow frequency, and suggest further analysis on channel junctions and concavity. These results pave the way to an implementation of a DFIM in 2D-LEMs.