  We consider the mean curvature flow of compact convex surfaces in Euclidean
$3$-space with free boundary lying on an arbitrary convex barrier surface with
bounded geometry. When the initial surface is sufficiently convex, depending
only on the geometry of the barrier, the flow contracts the surface to a point
in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere
lying in a half space. This extends, in dimension two, the convergence result
of Stahl for umbilic barriers to general convex barriers. We introduce a new
perturbation argument to establish fundamental convexity and pinching estimates
for the flow. Our result can be compared to a celebrated convergence theorem of
Huisken for mean curvature flow of convex hypersurfaces in Riemannian
manifolds.
