Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that $f|_S:S\to P$ is proper and $f|_X:X\to P$ is a submersion for any stratum $X$ of $S$, then $f|_S:S\to f(S)$ is a locally trivial fibration. Does this imply that $f|_X:X\to f(X)$ is a locally trivial fibration, for any stratum $X$ of $S$?
Yes. Notice that since $f|_S: S \to S$ is proper, so is $f|_X$ (because $X$ is closed). Furthermore, $f|_X$ is a submersion and is surjective onto its image. So by a direct application of Ehresmann's fibration lemma you get your result.
Also note that because any fibration is also a compactification of a subset of all strata, we can always divide any subsets of these strata into an infinite number of compactified stratified and homotopically self similar spaces. Also note that the isotopy from strata 1 to strata 2 is a homotopy as well.