The paper introduces a methodology for numerical simulation of landslides experiencing considerable deformations and topological changes. Within an interface capturing approach, all interfaces are implicitly described by specifically defined level-set functions allowing arbitrarily evolving complex topologies. The transient interface evolution is obtained by solving the level-set equation driven by the physical velocity field for all three level-set functions in a block Jacobi approach. The three boundary-coupled fluid-like continua involved are modeled as incompressible, governed by a generalized non-Newtonian material law taking into account capillary pressure at moving fluid–fluid interfaces. The weighted residual formulation of the level-set equations and the fluid equations is discretized with finite elements in space and time using velocity and pressure as unknown variables. Non-smooth solution characteristics are represented by enriched approximations to fluid velocity (weak discontinuity) and fluid pressure (strong discontinuity). Special attention is given to the construction of enriched approximations for elements containing evolving triple junctions. Numerical examples of three-fluid tank sloshing and air–water-liquefied soil landslides demonstrate the potential and applicability of the method in geotechnical investigations. Copyright © 2012 John Wiley & Sons, Ltd.