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Periocular biometric, the peripheral area of the ocular, is a collaborative alternative to the face, especially when the face is occluded or masked. However, in practice, sole periocular biometric capture the least salient facial features, thereby lacking discriminative information, particularly in wild environments. To address these problems, we transfer discriminatory information from the face to support the training of a periocular network by using knowledge distillation. Specifically, we leverage face images for periocular embedding learning, but periocular alone is utilized for identity identification or verification. To enhance periocular embeddings by face effectively, we proposeConsistent Knowledge Distillation (CKD) that imposes consistency between face and periocular networks across prediction and feature layers. We find that imposing consistency at the prediction layer enables (1) extraction of global discriminative relationship information from face images and (2) effective transfer of the information from the face network to the periocular network. Particularly, consistency regularizes the prediction units to extract and store profound inter-class relationship information of face images. (3) The feature layer consistency, on the other hand, makes the periocular features robust against identity-irrelevant attributes. Overall, CKD empowers the sole periocular network to produce robust discriminative embeddings for periocular recognition in the wild. We theoretically and empirically validate the core principles of the distillation mechanism in CKD, discovering that CKD is equivalent to label smoothing with a novel sparsity-oriented regularizer that helps the network prediction to capture the global discriminative relationship. Extensive experiments reveal that CKD achieves state-of-the-art results on standard periocular recognition benchmark datasets.