Group re-identification (GReID) aims to correctly associate group images belonging to the same group identity, which is a crucial task for video surveillance. Existing methods only model the member feature representations inside each image (regarded as spatial members), which leads to potential failures in long-term video surveillance due to cloth-changing behaviors. Therefore, we focus on a new task called cloth-changing group re-identification (CCGReID), which needs to consider group relationship modeling in GReID and robust group representation against cloth-changing members. In this paper, we propose the separable spatial-temporal residual graph (SSRG) for CCGReID. Unlike existing GReID methods, SSRG considers both spatial members inside each group image and temporal members among multiple group images with the same identity. Specifically, SSRG constructs full graphs for each group identity within the batched data, which will be completely and non-redundantly separated into the spatial member graph (SMG) and temporal member graph (TMG). SMG aims to extract group features from spatial members, and TMG improves the robustness of the cloth-changing members by feature propagation. The separability enables SSRG to be available in the inference rather than only assisting supervised training. The residual guarantees efficient SSRG learning for SMG and TMG. To expedite research in CCGReID, we develop two datasets, including GroupPRCC and GroupVC, based on the existing CCReID datasets. The experimental results show that SSRG achieves state-of-the-art performance, including the best accuracy and low degradation (only 2.15% on GroupVC). Moreover, SSRG can be well generalized to the GReID task. As a weakly supervised method, SSRG surpasses the performance of some supervised methods and even approaches the best performance on the CSG dataset.