Let $\pi:X\to\mathcal X$ be a presentation of an Artin stack $\mathcal X$ of finite type over a field $k,$ and let $f:Y\to X$ be a finite \'etale covering. Does there exist a finite \'etale covering $Y'\to X$ factoring through $Y,$ such that $Y'$ can be given descent structure, i.e. there exists an isomorphism $pr_1^*Y'\cong pr_2^*Y'$ over $X\times_{\mathcal X}X$ satisfying cocycle condition, so that $Y'\to X$ descend to a finite \'etale covering $\mathcal Y\to\mathcal X$?
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I don't think so (finite etale covers cannot be localized in smooth topology in the sense that you describe). Say, $\mathcal{X}$ is a point, and $X$ is a smooth variety with nontrivial fundamental group, say, an elliptic curve (or $\mathbb{A}^1\{0\}$). Then $\pi$ is a presentation. Let $f:Y\to X$ be a nontrivial finite etale cover, say, the cover of the elliptic curve by an isogeneous elliptic curve. Then your question becomes: `is there a trivial (i.e., lifted from $\mathcal{X}$) cover $Y'$ of $X$ with a map to $Y$? This is of course not true. 

