Sixth generation (6G) wireless networks are poised to revolutionize connectivity by integrating terrestrial, aerial, and satellite communication segments into a unified 3D architecture. However, this integration introduces unprecedented challenges in mobility management due to the dynamic and heterogeneous nature of the network. The high mobility of low earth orbit (LEO) satellites, unmanned aerial vehicles (UAVs), and ground users requires frequent and seamless handovers, complicating channel estimation, resource allocation, and routing. These challenges arise from the rapid relative motion between network entities, leading to time-varying channels, Doppler shifts, and unpredictable visibility windows. In addition, the diverse operational altitudes and energy constraints of UAVs further complicate trajectory planning and energy-efficient operation. This paper provides a comprehensive analysis of these mobility management challenges, exploring their underlying causes and importance in ensuring reliable, low-latency communication in 6G networks. We examine key techniques such as time-varying channel estimation, dynamic resource allocation, seamless handovers, energy-aware UAV trajectory planning, and 3D dynamic routing. A case study demonstrates joint optimization of the UAV trajectory and satellite handovers using online learning, showcasing the role of AI in addressing these issues. The paper concludes with a discussion on future directions, including the need for distributed mobility management architectures, mobility-aware modulation schemes, and proactive handover strategies, all critical for realizing the full potential of 6G networks.