An assembly is provided for a gas turbine engine. This assembly includes a multi-walled structure including a cold wall, a hot wall and a cooling cavity vertically between the cold wall and the hot wall. The cold wall includes a plurality of cold wall apertures fluidly coupled with the cooling cavity. The cold wall apertures are configured to subject the cold wall to a cold wall pressure drop vertically across the cold wall. The hot wall includes a plurality of hot wall apertures fluid coupled with the cooling cavity. The hot wall apertures are configured to subject the hot wall to a hot wall pressure drop vertically across the hot wall that is greater than or equal to the cold wall pressure drop.