More generally, two homeomorphisms of Dn that are isotopic on the boundary, are isotopic.
If satisfies , then an isotopy connecting f to the identity is given by
Visually, you straighten it out from the boundary, squeezing down to the origin. William Thurston calls this "combing all the tangles to one point".
The subtlety is that at , "disappears": the germ at the origin "jumps" from an infinitely stretched version of to the identity. Each of the steps in the homotopy could be smoothed (smooth the transition), but the homotopy (the overall map) has a singularity at . This underlines that the Alexander trick is a PL construction, but not smooth.
General case: isotopic on boundary implies isotopic
Now if are two homeomorphisms that agree on , then is the identity on , so we have an isotopy from the identity to . The map is then an isotopy from to .
However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true piecewise-linearly, but not smoothly.
Concretely, let be a homeomorphism, then