The presence of a free surface between two fluids of different properties is a source of (geometric) nonlinearity due to the role of interface curvature on the normal stress balance when surface tension is present. Thus, even for linear bulk equations (Stokes flow), a rich solution structure can be found. One difficulty in solving such problems is that the free surface and the fluid domain(s) can change dramatically with time, or under variation of parameters. We have developed numerical methods to solve such problems using an ALE-based finite-element method. The deforming fluid domain is treated as a pseudo-elastic solid for small deformations, but can also be completely remeshed to handle extreme changes in geometry. Techniques for the continuation of solution branches in the presence of remeshing will be described and used to demonstrate the existence of new solutions for the canonical problems of viscous fluid flow on the outside or inside of rotating cylinders, and to quantify the accuracy of the widely-used thin-film approximations in such flows. The same techniques will also be used to characterise the solution structure that develops for two-phase flow in partially-occluded Hele-Shaw cells. That system becomes more sensitive as the aspect ratio is increased in the sense that multiple solutions are provoked for smaller occlusions, which is conjectured to underlie the experimentally observed sensitivity of such cells at large aspect ratios.