Graphs are continually growing in size, and processing complex queries, such as multi-hop pattern queries, on them is becoming increasingly important. Although GPUs have received significant attention recently, there is still a notable shortage of efficient GPU-based out-of-memory methods for handling these queries. Three key issues arise when processing multi-hop queries on large-scale graphs using GPUs: the need for an efficient graph format, effective scheduling of accesses to graph partitions on storage, and dynamic buffer management on both the host and GPUs. To address these issues, we propose an efficient GPU-based out-of-memory multi-hop query processing framework called GFlux. Through extensive experiments, we have demonstrated that GFlux significantly improves both the speed and scalability compared to existing state-of-the-art methods.