This question is based on Lemma 3.3, page 6 in this paper: http://arxiv.org/pdf/1106.0622v4.pdf I changed the notation quite a lot, but it should be a one-to-one correspondence.
$S(x)$ is a compact manifold for each $x \in [0,T]$.
Fix $s \in [0,T]$. Let $t \in [0,T]$. Suppose $f(t,s):H^{-1}(S(s)) \to H^{-1}(S(t))$ (linear functional), with the property that $f(t,t)$ is the identity for any $t$, and suppose the following holds:
$\frac{1}{1+|s-t|}\lVert u\rVert_{H^{-1}(S(s))} \leq \lVert f(t,s)u \rVert_{H^{-1}(S(t))} \leq \frac{1}{1-|s-t|}\lVert u\rVert_{H^{-1}(S(s))}$
It might be helpful to know the adjoint of $f(s,t)$, written $f(s,t)^*:H^1(S(s)) \to H^1(S(t))$ has the property that $\lVert f(s,t)^*v \rVert_{H^1(S(t))}$ is continuous as a function of $t$.
The task is to show that $\lVert f(t,s)u \rVert_{H^{-1}(S(t))}$ is continuous as a function of $t$.
Clearly we can see that it is continuous at $t=s$. But how about apart from $s$? How to see that it is continuous?