Large Language Models (LLMs) often suffer from performance degradation when faced with domain shifts, primarily due to catastrophic forgetting. In this work, we propose KILO (Knowledge-Instructed Learning for Continual Adaptation), a novel continual learning framework that integrates dynamic knowledge graphs with instruction tuning. By leveraging retrieved domain-specific knowledge as guidance during training, KILO enhances both adaptability to new domains and retention of previously acquired knowledge. We pretrain our model on WikiText-103 and evaluate sequential adaptation across four diverse target domains: BioASQ, SciQ, TweetEval, and MIND. Our experiments demonstrate that KILO consistently outperforms strong baselines, including continual fine-tuning, ERNIE 2.0, and CPT, in terms of backward transfer, forward transfer, F1 score, retention rate, and training efficiency. These results highlight the effectiveness of combining structured knowledge retrieval and instruction prompting to overcome domain shift challenges in continual learning scenarios.