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Owing to the controlling flexibility and cost-effectiveness, fixed-wing unmanned aerial vehicles (UAVs) are expected to serve as flying base stations (BSs) in the air-ground integrated network. By exploiting the mobility of UAVs, controllable coverage can be provided for mobile group users (MGUs) under challenging scenarios or even somewhere without communication infrastructure. However, in such dual mobility scenario where the UAV and MGUs are all moving, both the non-hovering feature of the fixed-wing UAV and the movement of MGUs will exacerbate the dynamic changes of user scheduling, which eventually leads to the degradation of MGUs' quality-of-service (QoS). In this paper, we propose a fixed-wing UAV-enabled wireless network architecture to provide moving coverage for MGUs. In order to achieve fairness among MGUs, we maximize the minimum average throughput between all users by jointly optimizing the user scheduling, resource allocation, and UAV trajectory control under the constraints on users' QoS requirements, communication resources, and UAV trajectory switching. Considering the optimization problem is mixed-integer non-convex, we decompose it into three optimization subproblems. An efficient algorithm is proposed to solve these three subproblems alternately till the convergence is realized. Simulation results demonstrate that the proposed algorithm can significantly improve the minimum average throughput of MGUs.