The development of early-warning systems for natural hazards such as landslides and lava flows is crucial for protecting communities in mountainous and volcanic regions. Accurately capturing the dynamics of these phenomena—characterized by a wide range of mechanical behaviors and velocity regimes—poses significant modeling and computational challenges. In this work, we focus on the runout phase of fast landslides and explore extensions of the same computational framework to simulate lava flows. For landslides, we develop two second-order numerical schemes based on depth-integrated models: one for homogeneous flows (e.g., mudflows) and another for flows with strong solid-liquid interactions (e.g., debris flows). Both schemes are formulated as scalable modifications of the two-step staggered Taylor-Galerkin (TG2) method on hierarchical quadtree meshes, equipped with adaptive mesh refinement and domain partitioning to ensure computational efficiency. To discretize non-conservative terms, eventually leading to well-balancing, and to address numerical stiffness due to reaction terms, we incorporate a path-conservative formulation along with either second-order operator splitting or implicit-explicit Runge-Kutta (IMEX) time integration. Building upon this landslide framework, we extend the methodology to simulate lava flows by solving a modified system of shallow water equations that accounts for energy transport and point-source terms representing eruptive vents. This extension preserves the well-balancing property and maintains high parallel scaling efficiency. The framework is validated through a suite of synthetic benchmarks and real-world test cases from the literature, demonstrating its accuracy, robustness, and applicability across hazard types. Overall, our unified approach provides an efficient and flexible simulation tool for depth-integrated modeling of geophysical mass flows in complex physical and topographical settings.