Geophysical mass flows, such as snow, ice, and rock avalanches, as well as debris flows, pose significant threats to human life and infrastructure in mountainous regions, necessitating advanced numerical models for accurate hazard evaluation and mitigation. State-of-the-art modeling techniques rely on depth-averaged models that often use simplified and heuristic frictional rheologies derived from open-channel hydraulics. Recently, three-dimensional particle-based methods relying on elastoplasticity showed promising results, yet at a high computational cost. We bridge these two worlds by employing a computationally efficient depth-averaged material point method (DAMPM) with an elastoplastic approach. In the DAMPM framework, conservation equations are derived with shallow-water assumptions and solved in a hybrid Eulerian-Lagrangian form as in 3D MPM, allowing for large deformations. DAMPM was first proposed by Abe and Konagai (2016) and has later been applied to simulate the release of snow slab avalanches (Guillet et al. 2023) and flow-like landslides (Fois et al. 2024). In our work, we use a Drucker-Prager yield criterion and different basal friction models such as the Voellmy model and the μ(I) rheology, include curvature effects, and apply the model to generic and real-world topography by comparing the elastoplastic framework to the more classical lateral earth pressure approach. We examine the accuracy of the model and the convergence behavior by also simulating problems that have an analytical solution. Furthermore, we explore how compressibility, which can easily be implemented within the elastoplastic framework, influences the results. We expect this model to contribute to improved risk assessment in mountainous regions by reducing computational cost in mass movement modeling while maintaining accuracy, as verified on the basis of benchmark tests.