Satellite-terrestrial integrated networks (STNs) are considered a promising architecture for 6G networks due to their ability to provide ubiquitous, high-capacity coverage on a global scale by combining satellite and terrestrial network infrastructures. However, the complex network architecture, time-varying topology, and frequent inter-satellite connection handovers present significant challenges for developing efficient routing and service continuity in STNs. To overcome these challenges, we propose Dyna-STN, a dynamic discrete topology-oriented wide-area routing mechanism in this paper. Dyna-STN utilizes a dynamic discrete topology model to characterize the time-varying topology of satellite networks. Furthermore, a hierarchical framework is established within the management plane to implement Dyna-STN, which comprises a dynamic discrete topology management plane and a routing management plane. A virtual overlay network composed of fixed virtual nodes shields the dynamics of satellite networks, while the open shortest path first protocol (OSPF) is deployed within the virtual overlay network to exchange routing reachability information among virtual nodes. Additionally, Dyna-STN performs dynamic binding and service migration between different satellite entities at specific time slots, thereby maintaining the continuity of virtual node services. Extensive numerical results demonstrate that Dyna-STN outperforms several baseline schemes in terms of routing protocol performance, packet forwarding performance, and service continuity. Furthermore, Dyna-STN maintains stable performance as the network scale increases and supports reliable data transmission among terminal devices in STNs.