Vortex beam carrying orbital angular momentum (OAM), which features a helical phase front, has shown its potential applications in diverse areas, especially in free-space optical (FSO) communications. However, when generating vortex beams, the radial phase distribution is usually disregarded in previous reports. In this paper, by controlling the radial phase distribution, we propose a method for the generation of vortex beams with arbitrary convex trajectories. By using this method, we successfully generate vortex beams with different predesigned trajectories with high accuracy. Moreover, we also demonstrate the transmission of the radial phase-controlled vortex beams in FSO links for different scenarios in simulation. Firstly, we generate vortex beams with different OAM states (l=+1, + 3, and +6), which have the same ring diameter at the receiver side. Secondly, we generate three vortex beams (l=+3) with the same ring diameter at different transmission distances (z = 100 m, 200 m, and 300 m). Finally, by carefully controlling the radial phase of the vortex beam, we generate vortex beams that can almost keep the same ring diameter for a long distance. The proposed method for shaping the transmission trajectory of vortex beams may pave the way for more applications in OAM-based FSO communications.