Graph models many real-world data like social, transportation, biology, and communication data. Hence, graph traversal including multi-hop or graph-walking queries has been the key operation atop graph stores. However, since different graph traversals may touch different sets of vertices, it is hard or even impossible to have a one-size-fits-all graph partitioning algorithm that preserves access locality for various graph traversal workloads. Meanwhile, prior shard-based migration faces a dilemma such that coarse-grained migration may incur more migration overhead over increased locality benefits, while fine-grained migration usually requires excessive metadata and incurs non-trivial maintenance costs. We present Pragh, an efficient locality-preserving live graph migration scheme for graph stores in the form of key-value pairs. The key idea of Pragh is a split migration model that only migrates values physically while retaining keys in the initial location. This allows fine-grained migration while avoiding the need to maintain excessive metadata. Pragh integrates an RDMA-friendly location cache from DrTM-KV to provide fully-localized access to migrated data and further makes a novel reuse of the cache replacement policy for lightweight monitoring. Pragh further supports evolving graphs through a check-and-forward mechanism to resolve the conflict between updates and migration of graph data. Evaluations on an 8-node RDMA-capable cluster (100 Gbps) using a representative graph traversal benchmark show that Pragh can increase the throughput by up to 19× and decrease the median latency by up to 94%, thanks to split live migration that eliminates 97% remote accesses. A port of split live migration to Wukong shows up to 2.53× throughput improvement on representative workloads like LUBM-10240, thanks to a reduction of 88% remote accesses. This further confirms the effectiveness and generality of Pragh. Finally, though Pragh focuses on RDMA-based graph traversal, we show its generality by extending it to support graph traversals under traditional networking. Evaluations on the graph traversal benchmarks and graph query workloads on the same cluster but with 10 Gbps TCP/IP network further confirm its effectiveness without RDMA. Specifically, when evaluating on the LUBM-10240, Wukong-TCP with Pragh can achieve up to 1.87× throughput improvement with a 56% decrease in remote accesses.