  In this paper, we study the forced mean curvature flows and the prescribed
mean curvature equations of both graphs and level-sets with capillary-type
boundary conditions on a $C^3$ bounded domain, which is not necessarily convex.
We prove a priori gradient estimates locally Lipschitz in time. Under an
assumption on the forcing term, we prove that the gradient estimates are
globally Lipschitz in time. As a consequence, we obtain the existence theorem
of solutions. In our formulation, we recover the known results of the gradient
estimates on a strictly convex $C^3$ bounded domain. Next, we study the
associated eigenvalue problems for mean curvature flows of both graphs and
level-sets. We prove the large time behavior of the solutions of mean curvature
flows of graphs on a smooth bounded domain. Finally, we compute the asymptotic
speed of the solutions of level-set mean curvature flows and the large time
profile of level-sets in the radially symmetric case based on optimal control
formula. Examples arising in the radially symmetric case demonstrate that the
additional assumption on the forcing term is optimal.
