A cooling system for a turbomachine includes a vane having an outer diameter platform at one end and an inner diameter platform at another end opposite the outer diameter platform. The outer diameter platform includes outer diameter attachment structure operative to mount the vane to an outer diameter stationary structure. The inner diameter platform is operative to be disposed on an inner diameter stationary structure. The cooling system also includes an airfoil disposed between the outer diameter platform and the inner diameter platform and defining a first cross-passage. The first cross-passage defines a first outer diameter opening in the outer diameter platform. The first cross-passage defines a first inner diameter opening in the inner diameter platform such that a first leakage flow can pass through the vane from the inner diameter platform, through the airfoil, to the outer diameter platform, and into an outer diameter leakage path of the vane.