An assembly (30) is provided for a gas turbine engine. This assembly (30) includes a multi-walled structure (32) including a cold wall (34), a hot wall (36) and a cooling cavity (38) vertically between the cold wall (34) and the hot wall (36). The cold wall (34) includes a plurality of cold wall apertures (80) fluidly coupled with the cooling cavity (38). The cold wall apertures (80) are configured to subject the cold wall (34) to a cold wall pressure drop vertically across the cold wall (34). The hot wall (36) includes a plurality of hot wall apertures (82) fluidly coupled with the cooling cavity (38). The hot wall apertures (82) are configured to subject the hot wall (36) to a hot wall pressure drop vertically across the hot wall (36) that is greater than or equal to the cold wall pressure drop.
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