In the class of mean convex surfaces, the mean curvature flow provides a useful geometric tool, owing its power to the regularity and structure theory established by White and with subsequent developments by Haslhofer, Kleiner and Hershkovits. In joint work with Edelen, Haslhofer and Ivaki, we generalise this theory to the free boundary setting. There are significant analytic and geometric issues in the passage to free boundary that we will discuss in the talk.