Dissipative chemical systems operate outside of equilibrium, and hold potential to enable life-like behavior in synthetic matter, such as self-organization, motility, and dynamic switching between different states. Here, out-of-equilibrium self-organization is demonstrated at an air-water interface, enabled by amphiphile filaments that self-assemble from source droplets and tether to pivalic anhydride-based drain droplets, which are surrounded by a pivalic acid gradient due to their hydrolysis. The coupling of chemical gradients, self-assembly and Marangoni flow due to release and depletion of amphiphiles at the air-water interface generates a unique orbiting of drain droplets around the source droplet. This orbiting is proposed to be driven by the selective adhesion of filaments to the front of the moving drain, while filaments approaching the drain from behind are destabilized upon contact with the asymmetrical gradient of pivalic acid. The motion sustains itself to complete multiple rotations, ending when the depletion of amphiphiles at the drain, which drives the Marangoni flow towards the drain, becomes too weak to attract new filaments. Potential applications are foreseen in rearranging networks for dynamic transfer of chemical signals amongst interconnected droplets, and the implementation of dissipative chemical reactions in self-organizing systems as a strategy towards life-like behavior is highlighted.