In SGA4 Exposé IX 2.14, there is a statement that given a constructible sheaf $F$ on a qcqs scheme $X$, not only can we choose a constructible stratification $\{X_i\to X\}$ of $X$ such that the pullback of $F$ to each $X_i$ is locally constant constructible, we can in fact choose this stratification to be isotrivial, that is, such that there exists a finite étale cover $U_i\to X_i$ with the property that the pullback of $F$ to each $U_i$ is a constant sheaf.

The reference for this is IX.2.8.1, which tells us that a for a surjective locally finite presentation map of schemes $Y\to X$, there exists a constructible stratification $\{X_i\to X\}$ together with a finite étale map $U_i\to X_i$ and a finite locally free radiciel map $V_i\to U_i$ such that the pullback of $Y$ to each $V_i$ admits a section.

Since $F\to X$ is a finite presentation étale algebraic space over $X$, we can cover it by some étale affine $F^\prime \to F$, we can apply the theorem to show that the data above exist to construct sections, and since the map $V_i\to U_i$ is a universal homeomorphism, sections over $V_i$ and $U_i$ will be the same, so we can just consider the case where we have the data $U_i\to X_i\to X$.

But I'm not clear on how to show that this proves the claim. What I tried was some argument approximating the base by Noetherian schemes and finding some étale sheaf over some index that pulls back to $F$ on $X$. Moreover, using the stratification of a constructible sheaf, you can assume it's locally constant constructible and with all fibres nonempty. Then in the Noetherian case, you apply induction on the size of the fibres by finding a global section over some finite étale $U\to X$, then taking the subsheaf complementary to the global section over $U$ (which is still lcc) and applying the inductive hypothesis.

Is this the 'easy argument' that the text references, or is there something easier?

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    $\begingroup$ Hmm, my guess is that to go from locally constant to constant, one should apply 2.8.1 to a cover $X'_i \to X_i$ over which $\mathscr F$ becomes constant. After refining the stratification $\{X_i \to X\}$, this gives surjective morphisms $$Y''_i \stackrel{g_i}\to Y'_i \stackrel{h_i}\to X_i$$ such that $g_i$ is finite radicial and $h_i$ finite étale and $\mathscr F|_{Y''_i}$ is constant. Then VIII.1.1 tells you that $\mathscr F|_{Y'_i}$ is already constant. $\endgroup$ Nov 6, 2020 at 22:20
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    $\begingroup$ @R.vanDobbendeBruyn Ah, great. That seems to work perfectly! You should answer the question so I can accept your answer! $\endgroup$ Nov 7, 2020 at 0:57


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