Mechanical computations in multiphase domains raise numerous difficulties from the generation of the initial mesh to its adaptation throughout the simulation. All alternatives to mesh adaptation, such as level-set methods, have the well-known drawback of inducing volume conservation issues. In this paper, a moving mesh method is coupled to a topological mesh adaptation technique in order to track moving and deforming interfaces in multiphase simulations, with a robust control of mesh quality. Level-set functions are used as intermediaries to enhance the mesh adaptation technique with a volume conservation constraint, which is compatible both with implicit and with body-fitted interfaces. Results show that this method has the same advantage of permitting important displacements, deformations, and topological changes (coalescence of interfaces, for example) as a standard level-set method, while volume diffusion is drastically reduced