Let $M$ be a smooth manifold. Let $f:M\rightarrow M$ be a diffeomorphism which is smoothly isotopic to the identity. Let $X\subset M$ be a compact subset such that $f|_X = id_X$.
Under what conditions will there exist a smooth isotopy $F: M\times [0,1]\rightarrow M\times [0,1]$ from the identity map to $f$ such that for any $t$, $F|_{X\times \{ t\} }$ is the identity?
This is more of an extended comment than an answer:
Here's a very simple counter-example to the statement in general. Let $M=S^2$ and let $X$ be the disjoint union of two non-intersecting disks centered at the poles. Define $f$ as the identity along the disks and along the cylinder between them let $f$ be a non-trivial Dehn Twist. It's relatively easy to see how to isotope the whole map to the identity (just untwist the Dehn twist), and I'll leave it as an exercise to show it isn't isotopic to the identity by an isotopy that fixes the polar disks (Hint: such a map induces a map on the torus).