The present thesis deals with gravity-driven thin-film flows in various configurations. We study the late-time drainage of a fixed liquid volume on a saddle topography, where the concurrent convergent and divergent natures of the flow render the pure-drainage equation, accounting only for the driving tangential component of gravity, singular. In fact, a hydrostatic boundary layer regularises the flow in the vicinity of the saddle point. We characterise it and discuss its universality to all topographies, presenting saddle points. We investigate the linear instability of uniform film flows at an imposed flow rate over inclined soluble surfaces, believed to carve an array of rills in the direction of the flow, known as linear karren patterns. We obtain a closed-form local dispersion relation and numerically calculate a global one. In the realm of spatially distributed volumetric sources, we obtain an analytical solution to the free-surface Stokes equations on spheres and cylinders in the case of uniform vertical volumetric flux through the interface, modelling rainfall. We analyse the linear stability of film flows on an inclined plane sustained by rain or vapour condensation. We apply a weakly non-parallel asymptotic method that is in perfect agreement with global calculations, and predict the linearly most amplified frequency. We explore the parallels between all of these configurations: late-time drainage of a fixed liquid volume, film flow with an imposed flow rate, as well as different spatially distributed sources, such as rain or vapour condensation. We believe that the unified framework behind all of them holds untapped potential for applications in geomorphogenesis involving thin-film flows. Finally, we construct an advection-less model for erosion by dilution-driven dissolution and apply it to the evolution of symmetric soluble blocks under rainfall.