We propose a novel transfer learning framework for pathological image analysis, the Response-based Cross-task Knowledge Distillation (RCKD), which improves the performance of the model by pretraining it on a large unlabeled dataset guided by a high-performance teacher model. RCKD first pretrains a student model to predict the nuclei segmentation results of the teacher model for unlabeled pathological images, and then fine-tunes the pretrained model for the downstream tasks, such as organ cancer sub-type classification and cancer region segmentation, using relatively small target datasets. Unlike conventional knowledge distillation, RCKD does not require that the target tasks of the teacher and student models be the same. Moreover, unlike conventional transfer learning, RCKD can transfer knowledge between models with different architectures. In addition, we propose a lightweight architecture, the Convolutional neural network with Spatial Attention by Transformers (CSAT), for processing high-resolution pathological images with limited memory and computation. CSAT exhibited a top-1 accuracy of 78.6% on ImageNet with only 3M parameters and 1.08 G multiply-accumulate (MAC) operations. When pretrained by RCKD, CSAT exhibited average classification and segmentation accuracies of 94.2% and 0.673 mIoU on six pathological image datasets, which is 4% and 0.043 mIoU higher than EfficientNet-B0, and 7.4% and 0.006 mIoU higher than ConvNextV2-Atto pretrained on ImageNet, respectively.