A scale-dependent dynamic subgrid model based on Lagrangian time averaging is proposed and tested in large eddy simulations (LES) of high-Reynolds number boundary layer flows over homogeneous and heterogeneous rough surfaces. The model is based on the Lagrangian dynamic Smagorinsky model in which required averages are accumulated in time, following fluid trajectories of the resolved velocity field. The model allows for scale dependence of the coefficient by including a second test-filtering operation to determine how the coefficient changes as a function of scale. The model also uses the empirical observation that when scale dependence occurs (such as when the filter scale approaches the limits of the inertial range), the classic dynamic model yields the coefficient value appropriate for the test-filter scale. Validation tests in LES of high Reynolds number, rough wall, boundary layer flow are performed at various resolutions. Results are compared with other eddy-viscosity subgrid-scale models. Unlike the Smagorinsky-Lilly model with wall-damping (which is overdissipative) or the scale-invariant dynamic model (which is underdissipative), the scale-dependent Lagrangian dynamic model is shown to have good dissipation characteristics. The model is also tested against detailed atmospheric boundary layer data that include measurements of the response of the flow to abrupt transitions in wall roughness. For such flows over variable surfaces, the plane-averaged version of the dynamic model is not appropriate and the Lagrangian averaging is desirable. The simulated wall stress overshoot and relaxation after a jump in surface roughness and the velocity profiles at several downstream distances from the jump are compared to the experimental data. Results show that the dynamic Smagorinsky coefficient close to the wall is very sensitive to the underlying local surface roughness, thus justifying the use of the Lagrangian formulation. In addition, the Lagrangian formulation reproduces experimental data more accurately than the planar-averaged formulation in simulations over heterogeneous rough walls.