We investigate the fluid mechanics of cleaning viscous drops attached to a flat inclined surface using thin gravity-driven film flows. We focus on the case where the drop cannot be detached from the surface by the mechanical forces exerted by the cleaning fluid on the drop surface. The fluid in the drop dissolves into the cleaning film flow, which then transports it away. We present a mathematical model for the mass transfer of the viscous fluid from the droplet into the film flow. The model assumes that the droplet has a negligible impact on the film velocity. To assess the impact of the drop on the velocity of the cleaning fluid, we have developed a novel experimental technique based on particle image velocimetry. We find that at intermediate Reynolds number the streamwise velocity can be strongly affected by the presence of the droplet. We discuss this impact on the cleaning of the droplet. Using the dye attenuation technique, we also measure the convective mass transfer of some dye mixed into the droplet and diffusing into the falling film. We find that the total amount of dye in the droplet decreases exponentially in time.