Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of any stratum is a union of some other strata). Let $\mathcal{F}^\bullet\in D^b(Sh_X)$ is an object of the bounded derived category of sheaves of vector spaces on $X$ whose cohomology sheaves are constructible along the strata of the given stratification.

Let $Y$ be a 'nice' topological space. Let $f\colon Y \times [0,1]\to X$ be a continuous map such that for any $y\in Y$ the image $f(\{y\}\times [0,1])$ is contained in some stratum.

**Question.** Is it true that $f^*(\mathcal{F}^\bullet)|_{Y\times\{0\}}$ and $f^*(\mathcal{F}^\bullet)|_{Y\times\{1\}}$ are isomorphic in $D^b(Sh_Y)$?

Remark. If $\mathcal{F}^\bullet$ is a local system, then the answer is positive regardless of any stratification.

A precise statement, ideally a reference, would be helpful (if it exists).