Several of Earth’s intra-plate volcanic provinces are hard to reconcile with the mantle plume hypothesis. Instead, they exhibit characteristics that are more compatible with shallower processes that involve the interplay between uppermost mantle flow and the base of Earth’s heterogeneous lithosphere. The mechanisms most commonly invoked are edge-driven convection (EDC) and shear-driven upwelling (SDU), both of which act to focus upwelling flow and the associated decompression melting adjacent to steps in lithospheric thickness. In this study, we undertake a systematic numerical investigation, in both 2-D and 3-D, to quantify the sensitivity of EDC, SDU, and the associated melting to key controlling parameters. Our simulations demonstrate that the spatio-temporal characteristics of EDC are sensitive to the geometry and material properties of the lithospheric step, in addition to the magnitude and depth-dependence of upper mantle viscosity. These simulations also indicate that asthenospheric shear can either enhance or reduce upwelling velocities and the associated melting, depending upon the magnitude and orientation of flow relative to the lithospheric step. When combined, such sensitivities explain why step changes in lithospheric thickness, which are common along cratonic edges and passive margins, only produce volcanism at isolated points in space and time. Our predicted trends of melt production suggest that, in the absence of potential interactions with mantle plumes, EDC and SDU are viable mechanisms only for Earth’s shorter-lived, lower-volume intra-plate volcanic provinces.