  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. Further, 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.
