Real-world Temporal Knowledge Graphs keep growing with time and new entities and facts emerge continually, necessitating a model that can extrapolate to future timestamps and transfer knowledge for new components. Therefore, our work first dives into this more realistic issue, lifelong TKG reasoning, where existing methods can only address part of the challenges. Specifically, we formulate lifelong TKG reasoning as a temporal-path-based reinforcement learning (RL) framework. Then, we add temporal displacement into the action space of RL to extrapolate for the future and further propose a temporal-rule-based reward shaping to guide the training. To transfer and update knowledge, we design a new edge-aware message passing module, where the embeddings of new entities and edges are inductive. We conduct extensive experiments on three newly constructed benchmarks for lifelong TKG reasoning. Experimental results show the outperforming effectiveness of our model against all well-adapted baselines.