  In this paper we study the mean curvature flow of embedded disks with free
boundary on an embedded cylinder or generalised cone of revolution, called the
support hypersurface. We determine regions of the interior of the support
hypersurface such that initial data is driven to a curvature singularity in
finite time or exists for all time and converges to a minimal disk. We further
classify the type of the singularity. We additionally present applications of
these results to the uniqueness problem for minimal hypersurfaces with free
boundary on such suppport hypersurfaces; the results obtained this way do not
require a-priori any symmetry or topological restrictions.
