  Let $L_t:=\Delta_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential
manifold $M$ with possible boundary $\partial M$, where $\Delta_t$ is the
Laplacian induced by a time dependent metric $g_t$ differentiable in $t\in
[0,T_c)$. We first introduce the damp gradient operator, defined on the path
space with reference measure $\mathbb{P}$, the law of the (reflecting)
diffusion process generated by $L_t$ on the base manifold; then establish the
integration by parts formula for underlying directional derivatives and prove
the log-Sobolev inequality for the associated Dirichlet form, which is further
applied to the free path spaces; and finally, establish numbers of
transportation-cost inequalities associated to the uniform distance, which are
equivalent to the curvature lower bound and the convexity of the boundary.
