Fog computing brings computational services closer to users and connected devices, addressing the increasing need for real-time data processing in Internet of Things (IoT) environments. As IoT devices and users move around frequently, ensuring consistent service quality has become increasingly challenging. Mobility-induced service migration has been recognized as a key mechanism to address this challenge, wherein services are migrated dynamically to nearby fog nodes in response to user mobility, thereby preserving low latency and ensuring continuous application performance. This literature review explores key research from recent years on mobility-aware service migration in Fog computing. We examine techniques such as virtual machine (VM) migration, container-based migration, and service function chaining (SFC) to understand how they reduce latency, energy consumption and bandwidth usage. A taxonomy of migration strategies is presented, along with a comparative analysis based on latency, overhead, and applicability. We discuss specific solutions like proactive VM handoff, container live migration frameworks, and SFC reconfiguration, highlighting their strengths and trade-offs in various domains such as smart cities, vehicular networks, and healthcare. The paper concludes by identifying ongoing challenges such as mobility prediction, energy-efficiency, and security in federated environments and outlines future research directions. This review aims to guide researchers in creating reliable Fog computing environment that support real-time applications under user mobility.