The driven-dissipative nature of light-matter interaction inside a multimode, dye-filled microcavity makes it an ideal system to study nonequilibrium phenomena, such as transport. In this work, we investigate how light is efficiently transported inside such a microcavity, mediated by incoherent absorption and emission processes. In particular, we show that there exist two distinct regimes of transport, viz. conductive and localized, arising from the complex interplay between the thermalizing effect of the dye molecules and the nonequilibrium influence of driving and loss. The propagation of light in the conductive regime occurs when several localized cavity modes undergo dynamical phase transitions to a condensed, or lasing, state. Furthermore, we observe that, while such transport is robust for weak disorder in the cavity potential, strong disorder can lead to localization of light even under good thermalizing conditions. Importantly, the exhibited transport and localization of light is a manifestation of the nonequilibrium dynamics rather than any coherent interference in the system.