According to the classical theory of film drainage for drops with tangentially immobile interfaces, the ambient flow field has no influence on film drainage aside from the hydrodynamic pushing force that it provides. The current theories of drop coalescence assume that this feature applies also to drops with tangentially mobile interfaces. This assumption is incorrect, except at short times. At long times, ambient flow conditions qualitatively affect film drainage, even under small-deformation conditions. The ambient flow can arrest film drainage, preventing coalescence, or exponentially enhance film drainage. Here, a new axisymmetric thin-film formulation is presented that correctly accounts for the influence of the ambient flow field. The formulation correctly recovers theoretical predictions for the stationary film thickness at long times. An analysis of the long-time behavior under accelerated drainage conditions explains why film thinning occurs exponentially. The numerical predictions of the new thin-film formulation are compared to whole-drop boundary integral simulations and are found to be in good agreement under small-deformation conditions. Thin-film simulations are used to describe offset collisions of drops, under the simplifying assumption that film drainage is unaffected by the tangential relative motion of the drops.