Homotopy property of constructible sheaves on stratified spaces

Here are two comments:

1) I suspect the answer is "yes," so long as your homotopy has the property that your pullback is locally constant (and hence, by triviality of the interval, constant) along the homotopy parameter, which I believe your condition implies. You might also want to add the hypothesis that the homotopy is stratified appropriately so that you can compare these sheaves against a common refinement.

2) Although this doesn't directly answer your question, I think your type of question is probably easier to understand with the equivalence between constructible sheaves and functors from the exit path category in mind.

The exit path category has points for objects and homotopy classes of exits paths for morphisms. An exit path is a continuous map from the interval with the property that the dimension of the containing stratum is non-decreasing. Homotopies are through exit paths.

Here is a short treatment of this equivalence done in the language of constructible cosheaves, although everything dualizes to the classical statement:

http://arxiv.org/abs/1603.01587

However, we abstract away the question whether every homotopy can be stratified and broken up into elementary homotopies.

We also give a new definition of "constructible," which I believe is better for many reasons.

As a caveat, I've heard it said that the derived category of constructible sheaves (take the category of constructible sheaves, then derive it) is not the same as the category of constructible derived sheaves (derived complexes of sheaves with constructible cohomology sheaves), but a theorem of Beilinson says they are the same for a triangulation.