An assembly is provided for a turbine engine. This turbine engine assembly includes a combustor wall with a shell and a heat shield. The combustor wall defines a quench aperture therethrough. The combustor wall also defines a cavity between the shell and the heat shield. The shell defines a first aperture through which air is directed into the cavity. The heat shield includes a rail that at least partially defines a second aperture configured to direct at least some of the air within the cavity out of the combustor wall and towards the quench aperture.