Consider a smooth ($C^\infty$) manifold $M$ and a family of Riemannian metrics $\{g^{(u)}\}$ on $M$ that depends smoothly on $u\in(a,b)$, in the sense that in certain coordinates $x^i$ on $M$ we can write
$$ g^{(u)} = g_{ij}(u,x)\,\text{d}x^i\text{d}x^j $$
for smooth components functions $g_{ij}$.
Now the situation is as follows. We are given some other metric $h_{ij}(x)$ such that for each fixed $u=u_0$, the metric $g_{ij}(u_0,x)$ can be transformed into $h_{ij}(x)$
My question is if these transformations can be chosen such that they depend smoothly on $u$ as well, i.e., does there exist a family of coordinate transformations $\phi_u:x\mapsto \tilde x$, $u\in (a,b)$ having the property that they transform $g_{ij}(u,x)$ into $h_{ij}(x)$, such that the map
$$ \phi: (u,x)\mapsto (u,\phi_u(x))$$
is smooth, at least locally?
EDIT: Let me give an example.
Let
$$g_{ij}(u,x^1,x^2) = \begin{pmatrix}1 & 0 \\ 0 & f(u)^2\end{pmatrix},\qquad f(u)\neq 0.$$
Clearly, for each $u$, there is a diffeomorphism $\phi_u(x^1,x^2) = (x^1,f(u)x^2)\equiv (y^1,y^2)$ such that in the new coordinates we have
$$\tilde g_{ij}(u,y^1,y^2) = \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}.$$
In this case the diffeomorphism depends smoothly on $u$ as well, in the sense that the map $\phi$, as defined above, now reads $\phi(u,x^1,x^2) = (u,x^1,f(u)x^2)$, which is a smooth function of $u$ provided that $f$ is smooth. In this example everything is very clear, but I'm wondering if such a construction is possible in general.
EDIT: Apparently the result follows from Ebin's slice theorem (see the answer below) but I don't see how. If someone could explain this that would be greatly appreciated. It also seems to me that the slice theory only applies to compact manifolds.
When you say "can be transformed" what you really mean is "isometric to".
A better way to state your question is as follows. Consider a smooth manifold $M$ (this will be a small coordinate neighborhood on your manifold), a smooth family of Riemannian metrics $g_u, u\in (0,1)$, on $M$, which are all isometric to a reference metric $h$. Does there exist a smooth family of isometries $f_u: (M, g_u)\to (M,h)$?
This is nontrivial, but the existence is due to
D. Ebin, The manifold of Riemannian metrics. 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) pp. 11–40, Amer. Math. Soc., Providence, R.I.
See Theorem 7.1 in this paper (a slice theorem). It may take you some time to translate from what he proves to what you want since he proves a much stronger result.