Abstract-In Cognitive Radio Networks (CRNs), Secondary Users (SUs) can flexibly access Primary Users' (PUs') idle spectrum bands but such spectrum opportunities are dynamic due to PUs' uncertain activity patterns. In a multi-hop CRN consisting of SUs as relays, such spectrum dynamics will further cause the invalidity of pre-determined routes. In this paper, we investigate spectrum-mobility-incurred route-switching problems in both spatial and frequency domains for CRNs, where spatial switching determines which relays and links should be re-selected and frequency switching decides which channels ought to be re-assigned to the spatial routes. The proposed route-switching scheme not only avoids conflicts with PUs but also mitigates spectrum congestion. Meanwhile, tradeoffs between routing costs and channel switching costs are achieved. We further formulate the route-switching problem as the Route-Switching Game which is shown to be a potential game and has a pure Nash Equilibrium (NE). Accordingly, efficient algorithms for finding the NE and the ϵ−NE are proposed. Then we extend the proposed game to the incomplete-information scenario and provide a method to compute the Bayesian NE. Finally, we prove that the price of stability of the proposed game has a deterministic upper bound.