This paper presents a hybrid framework for literature reviews that augments traditional bibliometric methods with large language models (LLMs). By fine-tuning open-source LLMs, our approach enables scalable extraction of qualitative insights from large volumes of research content, enhancing both the breadth and depth of knowledge synthesis. To improve annotation efficiency and consistency, we introduce an error-focused validation process in which LLMs generate initial labels and human reviewers correct misclassifications. Applying this framework to over 20000 scientific articles about human migration, we demonstrate that a domain-adapted LLM can serve as a "specialist" model - capable of accurately selecting relevant studies, detecting emerging trends, and identifying critical research gaps. Notably, the LLM-assisted review reveals a growing scholarly interest in climate-induced migration. However, existing literature disproportionately centers on a narrow set of environmental hazards (e.g., floods, droughts, sea-level rise, and land degradation), while overlooking others that more directly affect human health and well-being, such as air and water pollution or infectious diseases. This imbalance highlights the need for more comprehensive research that goes beyond physical environmental changes to examine their ecological and societal consequences, particularly in shaping migration as an adaptive response. Overall, our proposed framework demonstrates the potential of fine-tuned LLMs to conduct more efficient, consistent, and insightful literature reviews across disciplines, ultimately accelerating knowledge synthesis and scientific discovery.