Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$.

Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a neighborhood of $\infty$?

Basically, to move one compact set to another, you don't need to radically deform the whole space. I was trying to prove it using the Annulus Theorem, and reduced it to this question:

If $J$ is a Jordan curve and $D$ is an open disc containing the trace of $J$ under some isotopy $G$ on $J$, then $G$ can be extended to $\overline{D}$ such that it's fixed on $\partial D$.

In other words, you'll have a Jordan curve sliding around inside a fixed, larger circle and want to extend it to the larger disc in a well-behaved way.

Does anyone know a reference or proof of this? If you can extend it in the annular region, then that's sufficient; you can concoct the extension inside $J$ from that. I've been stuck on it for a few days, now; I know how to prove it when $F$ is smooth, but I don't know how to use that to get the topological version.

Thanks!