Flow of incompressible Newtonian liquid films is governed by the Navier–Stokes system with shear‐free, balanced‐normal‐stress, and kinematic boundary conditions at the free surface. This system is solved here for the evolution of finite‐amplitude two‐dimensional disturbances to otherwise steady flow down a vertical plate by means of a finite element method adapted for free boundary problems. When flow is specified to bespatiallyperiodic, fully developed steady flows that ensue approach time‐periodic states, i.e., waves, the finite amplitude of which depends upon their wavelength. The family of time‐periodic states connects to the steady, fully developed flow at a Hopf bifurcation that lies at a critical disturbance length, in agreement with the Orr–Sommerfeld analysis. Initial disturbances to flow down a plate offinitelengthgrow as they propagate downward. In all cases studied here, however, steady flow is eventually approached.