Surface-facilitated, front-propagated softening of glassy materials is now a well-known phenomenon, which is common to stable vapor deposited glasses. As we demonstrate in our recent communication, this softening pathway is not unique to vapor deposited vitreous phases and can be observed in ordinary melt-cooled glasses in the limit of high heating rates [Cubeta et al., J. Chem. Phys. 147(7), 071101 (2017)]. Expanding on this preliminary report, we use our thin-wire, quasi-adiabatic fast scanning calorimetry technique to investigate softening kinetics of micrometer scale, viscous liquid methylbenzene, and 2-propanol films, which are fully equilibrated at distinct temperatures near the compounds' standard glass hardening transition ranges. Heating of each sample with rates in excess of 105 K.s(-1) results in softening kinetics that are well approximated by an Arrhenius temperature function. Remarkably, the apparent activation energy barriers to non-equilibrium, front-propagated softening matches the barriers to near-equilibrium self-diffusivity at the samples' initial temperatures. Furthermore, our analysis also shows an exceptionally strong correlation between the high temperature softening rate and the self-diffusion coefficients at low initial temperatures. Finally, our front softening velocities are also strongly dependent on the samples' initial states, much more so than previously observed. Based on these results, we propose an extended Wilson-Frenkel model of non-equilibrium phase transformations as a general theoretical framework to describe front propagated softening in glassy materials. Published under license by AIP Publishing.