Intelligent reflecting surface (IRS) can reconfigure the channel conditions, while the passive beamforming gain is limited by the severe double path-loss effect. Fortunately, active IRS (AIRS) is emerging to tackle obstacles by simultaneously adjusting the phase and amplitude of each reflection element. In this paper, we propose an AIRS-assisted unmanned aerial vehicle (UAV)-relaying scheme, where the AIRS is equipped on the UAV to reflect the signal from the ground base station (GBS) to users via non-orthogonal multiple access. We jointly adjust beamforming vectors at the GBS, reflection matrix of the AIRS and UAV trajectory to maximize the average sum rate. However, the problem is non-convex. Thus, it is decomposed into three subproblems via block coordinate descent. The beamforming optimization at the GBS is transformed into a standard semidefinite program through semidefinite relaxation. Then, the reflection matrix of AIRS and UAV trajectory subproblems are solved through successive convex approximation. Ultimately, we design an iterative algorithm to effectively tackle the original problem. Simulation results are shown to verify the performance of designed scheme.