Due to complicated situations such as node/link interference and traffic load, quality of service support in multi-hop multi-rate ad hoc networks remains a challenging issue. Furthermore, when mobility is present, because of frequent route change, it is even more difficult to maintain high level performance for existing real-time flows that may not tolerate serious performance degradation. In this paper, we focus on the issue of providing sufficient QoS in networks with moderate to high node mobility. We first introduce route available bandwidth (RAB), a major index for prediction of flow performance along a specific path. RAB considers various important factors such as intra-flow interference, effective link bandwidth, and channel busy time. Then, we devise a DSR based admission control scheme where the source node accepts/rejects a flow according to the RAB of the collected route when it receives a route reply packet. To handle mobility, we incorporate an additional metric of route reliability (RR) so that a path with sufficient bandwidth and reliability can be obtained. Results show that with admission control, existing flows experiences around 80-95% average delivery ratio, even for 20 m/s maximum moving speed, which is much better than 30-60% average delivery ratio when admission control is absent.