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Lifelong person re-identification (LReID) is in significant demand for real-world development as a large amount of ReID data is captured from diverse locations over time and cannot be accessed at once inherently. However, a key challenge for LReID is how to incrementally preserve old knowledge and gradually add new capabilities to the system. Unlike most existing LReID methods, which mainly focus on dealing with catastrophic forgetting, our focus is on a more challenging problem, which is, not only trying to reduce the forgetting on old tasks but also aiming to improve the model performance on both new and old tasks during the lifelong learning process. Inspired by the biological process of human cognition where the somatosensory neocortex and the hippocampus work together in memory consolidation, we formulated a model called Knowledge Refreshing and Consolidation (KRC) that achieves both positive forward and backward transfer. More specifically, a knowledge refreshing scheme is incorporated with the knowledge rehearsal mechanism to enable bi-directional knowledge transfer by introducing a dynamic memory model and an adaptive working model. Moreover, a knowledge consolidation scheme operating on the dual space further improves model stability over the long term. Extensive evaluations show KRC's superiority over the state-of-the-art LReID methods on challenging pedestrian benchmarks.