Remaining useful life (RUL) prediction is critical for improving system reliability and reducing maintenance costs. However, significant discrepancies in feature distributions across different degradation stages hinder traditional methods from accurately identifying stage transitions. Moreover, during continual learning across degradation stages, models are prone to catastrophic forgetting, which undermines their prediction stability and generalization capability. To address these challenges, this article proposes a degradation-aware continual learning approach for cross-stage RUL prediction. First, a baseline guided gradient offset detection method was developed based on a constructed health indicator that characterizes degradation trends, which divides the training data into multiple degradation stages. Then, a stage aware residual neural network with enhanced residual convolutional network is trained on multistage data, which adaptively identifies the current degradation stage during online testing. Finally, to address the issues of cross-stage feature modeling and catastrophic forgetting, this article designs a cross-stage RUL prediction model based on the memory-aware adaptive transformation network, which effectively enhances the model’s ability to represent stage features and improves its continual learning performance. Experimental results on both battery and XJTU-SY datasets demonstrate the superiority of the proposed method in RUL prediction, highlighting its strong generalization ability and practical applicability.