Let $X$ be a scheme over a field $k$ of characteristic $0$ and $\mathcal F$ is a torsion sheaf on $X$. Let $K/k$ be a purely transcendental extension. Is the natural map $H^i_{et}(X, \mathcal F) \to H^i_{et}(X_K, \mathcal F_K)$ an isomorphism?
An attempt: Suppose $K$ has transcendence degree $n$ over $k$. We can write the natural map $Spec (K) \to Spec (k)$ as a limit of smooth morphisms $U \to Spec(k)$, where $U$ varies over open subschemes of $\mathbb A^n$. Then the smooth base change theorem in etale cohomology gives the above result. Is this a correct argument?
Note that in this setting you can't even speak of $\mathcal F_K$. Indeed, $\mathcal F$ is only defined on the small étale site, and the morphism $\operatorname{Spec} K \to \operatorname{Spec} k$ is not in the small site.
But even if $\mathcal F$ is defined on the big étale site, so that the notation $\mathcal F_K$ makes sense, the result is false! Namely, let $k$ be algebraically closed, $K = k(t)$, and $X = \operatorname{Spec} k$. Let $\mathcal F = \mu_n$, and $i = 1$. Then the Kummer sequence (plus Hilbert 90) gives $H^1_{\operatorname{ét}}(X,\mathcal F) = k^\times/(k^\times)^n$, which is $0$ since $\mathcal k$ is algebraically closed. On the other hand, not every element of $K$ is an $n$-th power (e.g. $t$), so $H^1_{\operatorname{ét}}(X_K, \mathcal F_K) \neq 0$.