An assembly is provided for a gas turbine engine. This assembly includes a combustor wall, a case and an inlet. The combustor wall is configured with a quench aperture. The case is displaced from the combustor wall such that a plenum is formed by and extends between the combustor wall and the case. The case includes a case wall, a deflector and a conduit. The deflector projects out from the case wall into the plenum towards the combustor wall. The deflector is arranged upstream of the quench aperture. The inlet to the conduit is arranged next to and downstream of the deflector.