We studied the interaction of atmospheric-pressure plasma jets of Ar or air with liquid films of an aliphatic hydrocarbon on moving solid substrates. The hydrodynamic jet-liquid interaction induces a track of lower film thickness. The chemical plasma-surface interaction oxidizes the liquid, leading to a local increase of the surface tension and a self-organized redistribution of the liquid film. We developed a numerical model that qualitatively reproduces the formation, instability and coarsening of the flow patterns observed in the experiments. Monitoring the liquid flow has potential as an in-situ, spatially and temporally resolved, diagnostic tool for the plasma-liquid surface interaction.