While the passage of a plume beneath a thin oceanic lithosphere is usually connected to a hot spot track, and thus can easily be traced back in time, the interaction between a plume and thick continental or cratonic lithosphere is less obvious. Although volcanic eruptions are a common feature for plumes reaching the surface beneath continents, especially in the form of large igneous provinces associated with the arrival of a plume head, it is unlikely that the entire plume track is marked by extrusive volcanism. The thickness of continental lithosphere, and especially cratonic lithosphere, may prohibit the eruption of magma in many places, and can re-focus extrusive volcanism towards places where the lithosphere is thinner or more permeable due to preexisting fault structures. However, even though no magma might be erupted, the passage of a continent over a hot mantle upwelling will be visible in the surface heat flux, even millions of years after the plume passage. In this study, we use numerical models to investigate how a plume passing beneath continental or cratonic lithosphere affects surface heat flux over time, and which parameters of the plume and subsurface structure are the most relevant for determining the size of the respective heat flux anomaly. We show that any kind of surface heat flux anomaly is associated with an erosional thinning of the base of the lithosphere, and greater thinning leads to larger heat flux anomalies. While the maximum of lithosphere thinning is observed at a position and time a few million years after the plate passes over the plume, heat flux anomalies related to conduction continue to increase, reaching a maximum about 80-150 Myr after passage over the plume. In the case of stagnant (stationary) plates, the delays between lithosphere thinning and heat flux anomaly are smaller and the observed anomalies are larger. Amplitudes of both thinning and heat flux anomalies are most sensitive to the viscosity of the asthenosphere and the lower lithosphere, because a lower viscosity facilitates basal erosion and thus increases heat fluxes. Also important are the interaction time between plume and plate, i.e. plate velocity or plume life time, and the plume strength / excess temperature. These results have important implications for understanding plume-lithosphere interactions in polar settings, e.g. Greenland and Antartica, and for various places in Africa, North America and China.