Knowledge distillation transfers knowledge from large models into small models, and has recently made remarkable achievements. However, few studies has investigated the mechanism of knowledge distillation against distribution shift. Distribution shift refers to the data distribution drifts between training and testing phases. In this paper, we reconsider the paradigm of knowledge distillation by reformulating the objective function in shift situations. Under the real scenarios, we propose a unified and systematic framework to benchmark knowledge distillation against two general distributional shifts including diversity and correlation shift. The evaluation benchmark covers more than 30 methods from algorithmic, data-driven, and optimization perspectives for five benchmark datasets. Overall, we conduct extensive experiments on the student model. We reveal intriguing observations of poor teaching performance under distribution shifts; in particular, complex algorithms and data augmentation offer limited gains in many cases.