High density direct currents (DC) are used to drive flux quanta via the Lorentz force towards a highly ordered "free flux flow" (FFF) dynamic state, made possible by the weak-pinning environment of high-quality, single-crystal samples of two low-Tc superconducting compounds, V3Si and LuNi2B2C. We report the effect of the magnetic field-dependent fluxon core size on flux flow resistivity rho_f. Much progress has been made in minimizing the technical challenges associated with the use of high currents. Attainment of a FFF phase is indicated by the saturation at highest currents of flux-flow dissipation levels that are well below the normal state resistance and have field-dependent values. The field dependence of the corresponding rho_f is shown to be consistent with a prediction based on a model for the decrease of flux core size at higher fields in weak-coupling BCS s-wave materials.