The deformation of a nonlinear pulse traveling in a dispersive random medium can be studied with asymptotic analysis based on separation of scales when the propagation distance is large compared to the correlation length of the random medium. We consider shallow water waves with a spatially random depth. We use a formulation in terms of a terrain-following Boussinesq system. We compute the effective evolution equation for the front pulse which can be written as a dissipative Kortweg-de Vries equation. We study the soliton dynamics driven by this system. We show, both theoretically and numerically, that a solitary wave is more robust than a linear wave in the early steps of the propagation. However, it eventually decays much faster after a critical distance corresponding to the loss of about half of its initial amplitude. We also perform an asymptotic analysis for a class of random bottom topographies. A universal behavior is captured through the asymptotic analysis of the metric term for the corresponding change to terrain-following coordinates. Within this class we characterize the effective height for highly disordered topographies. The probabilistic asymptotic results are illustrated by performing Monte Carlo simulations with a Schwarz–Christoffel Toolbox.