Convexity estimates for mean curvature flow with free boundary

We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier $S$. Here $S$ can be any properly embedded, oriented surface in $R^{n+1}$ of bounded geometry. We also give an alternative proof that convex mean curvature flows with free boundary in $S^n$ pinch to umbilic.