Knowledge Distillation (KD) has been a popular paradigm for training a (smaller) student model from its teacher model. However, little research has been done on the practical scenario where only a subset of the teacher's knowledge needs to be distilled, which we term selective KD (SelKD). This demand is especially pronounced in the era of foundation models, where the teacher model can be significantly larger than the student model. To address this issue, we propose to rethink the knowledge distillation problem from the perspective of Inverse Optimal Transport (IOT). Previous Bayesian frameworks mapped each sample to the probabilities of corresponding labels in an end-to-end manner, which fixed the number of classification categories and hindered effective partial knowledge transfer. In contrast, IOT calculates from the standpoint of transportation or matching, allowing for the flexible selection of samples and their quantities for matching. Traditional logit-based KD can be viewed as a special case within the IOT framework. Building on this IOT foundation, we formalize this setting in the context of classification, where only selected categories from the teacher's category space are required to be recognized by the student in the context of closed-set recognition, which we call closed-set SelKD, enhancing the student's performance on specific subtasks. Furthermore, we extend the closed-set SelKD, introducing an open-set version of SelKD, where the student model is required to provide a "not selected" response for categories outside its assigned task. Experimental results on standard benchmarks demonstrate the superiority of our approach. The source code is available at: \href{https://github.com/machoshi/SelKD}