Thermo-mechanical thinning of the lithosphere by mantle plumes is essential for intra-plate volcanism, the initiation of rifting, the evolution of Earth’s lower continental crust and the genesis of metals, diamonds and hydrocarbons. To develop a new understanding of how a mantle plume thins the overlying lithosphere beneath moving plates, we use 2-D and 3-D numerical models based on a finite-element discretization on anisotropic adaptive meshes. Our models include Earth-like material properties for the upper mantle (e.g. temperature and viscosity contrasts, non-Newtonian rheology) discretised at a local mesh resolution that has previously been considered intractable. In our simulations, a plume is injected at the base of the model (670 km depth) with a prescribed mass flux that is consistent with surface observations of topographic swells: from 0.5 (e.g. Louisville, Bermuda, Darfur) to 7 Mg/s (Hawaii). We undertake a systematic numerical study, across a wide parameter space, to investigate the effect of plume buoyancy flux, plate velocity, rheology law and Rayleigh number on processes leading to a reduction of the depth of the Lithosphere Asthenosphere boundary (LAB), such as small-scale convection (SSC) (‘dripping’), or delamination of the lower lithosphere.