I'm currently reading through Kock's notes on TQFTs, found here http://mat.uab.es/~kock/TQFT/FS.pdf, and I assume all of his definitions are standard.
Suppose we have two diffeomorphic, closed, compact, smooth $(n-1)$-manifolds $\Sigma_0$ and $\Sigma_1$, with $\psi_1, \psi_2: \Sigma_0 \to \Sigma_1$ being diffeomorphisms. These diffeomorphisms induce cobordisms from $\Sigma_0$ to $\Sigma_1$ via the cylinder construction. (1.1.4 in the notes)
If $\psi_1$ and $\psi_2$ are smoothly homotopic, then we have a smooth map $F: \Sigma_0 \times I \to \Sigma_1$, where $F(x,0) = \psi_1(x)$ and $F(x,1) = \psi_2(x)$ for all $x \in \Sigma_0$. There is an obvious map $\Sigma_0 \times I \to \Sigma_1 \times I$ given by $(x,t) \mapsto (F(x,t),t)$. The notes claim that this is an equivalence of cobordisms from $\Sigma_0$ to $\Sigma_1$, after appropriate identifications of the boundary to make them actual cobordisms from $\Sigma_0$ to $\Sigma_1$.
However, I don't think $(x,t) \mapsto (F(x,t),t)$ is a diffeomorphism, unless $F_t$ is a diffeomorphism for all $t$, which I haven't assumed. Must I assume this?
Edit: The homotopy need not be through diffeomorphisms (isotopy) but the homotopy map is not a priori guaranteed to be a diffeomorphism.