Edge computing has become critical for enabling latency-sensitive applications, especially when paired with cloud resources to form cloud-assisted edge clusters. However, efficient resource management remains challenging due to edge nodes' limited capacity and unreliable connectivity. This paper introduces KubeDSM, a Kubernetes-based dynamic scheduling and migration framework tailored for cloud-assisted edge environments. KubeDSM addresses the challenges of resource fragmentation, dynamic scheduling, and live migration while ensuring Quality of Service (QoS) for latency-sensitive applications. Unlike Kubernetes' default scheduler, KubeDSM adopts batch scheduling to minimize resource fragmentation and incorporates a live migration mechanism to optimize edge resource utilization. Specifically, KubeDSM facilitates three key operations: intra-edge migration to reduce fragmentation, edge-to-cloud migration during resource shortages, and cloud-to-edge migration when resources become available, thereby increasing the number of pods allocated to the edge. Our results demonstrate that KubeDSM consistently achieves a higher average edge ratio and a lower standard deviation in edge ratios, highlighting its ability to provide more effective and stable scheduling across different deployments. We also explore the impact of migration strategies and Quality of Service (QoS) configurations on the edge ratios achieved by KubeDSM. The findings reveal that enabling migrations significantly enhances the edge ratio by reducing fragmentation. Additionally, KubeDSM's adaptability in respecting QoS requirements while maximizing overall edge ratios is confirmed through different QoS scenarios.