Flexible barriers have been increasingly used worldwide for mitigating natural hazards involving various geophysical flows. The analysis and design of flexible barriers need to distinguish two major impact mechanisms, runup and pile-up, which remain poorly understood due to complicated interactions between the impacting flow and the deformable barrier. In this study, a unified computational approach based on coupled computational fluid dynamics and discrete element method (CFD-DEM) is employed to examine the impact mechanisms of a wide spectrum of geophysical flows against a deformable and permeable flexible barrier. We consider geophysical flows including rock avalanche, debris avalanche, debris flow, debris flood and mud flow, and a flexible barrier consisting of a barrier net, cables and brakes. The signatures of runup and pile-up mechanisms are thoroughly analyzed, in terms of flow features and barrier responses. The effects of Froude numbers (Fr = 0.46-7.40), solid volume concentrations (0.1-1), and fluid rheologies (Newtonian and non-Newtonian fluids) on the impact mechanism transitions are systematically examined. Two nondimensional indices, the static-peak load ratio of the barrier and the momentum reduction ratio of the flow, are proposed for identifying the transition of impact mechanism from pile-up to runup. The transition is found to occur with either an increase in Fr or solid volume concentration of the flow. Increased viscosity of the fluid in a solid-fluid mixture may result in a transition occurring at a higher Fr. This study helps gain insights into the two impact mechanisms and their transitions and may provide a useful reference for the future design of flexible barriers in mitigating hazardous geophysical flows.