@AriyanJavanpeykar I agree this should be true if $U$ is a Zariski open. Does it hold for all opens? Take $U$ to be the open unit disc inside $X=\mathbf{P}^1$. Then every map $f:D \to X$ sending $D^*$ into $U$ sends the origin into $U$ (since the image of $D\to X$ is open (if $f$ is not constant) and contained in $U\cup \{f(0)\}$). But $U$ is not compact.

Does the valuative criterion work as stated for complex spaces? I’m afraid that essential singularities will ruin the analogy with algebraic geometry. For example, consider the map $\mathbb{D}^*\to \mathbf{P}^1$ given by $(e^{1/z}:1)$. It doesn’t extend across the puncture since the function is not meromorphic at zero.

However, things should be OK if you consider only $D_X$-modules which are "locally constant" on a given Zariski stratification of $X$.

Not every holonomic $D_U$-module extends to $X$: let $X = \mathbf{C}$, $U=\mathbf{C}\setminus 0$, let $Z\subseteq U$ be a discrete subset with point of accumulation $0$. Then the skyscraper $D_U$-module supported on $Z$ ($i_+ \mathcal{O}_Z$ where $i\colon Z\to U$ is the inclusion) is holonomic but does not extend to a coherent $D_X$-module.

Keerthi's comment is spot on and IMO should be posted as an answer.

Intuition, perhaps misguided: Aren't you secretly choosing two embeddings $\overline{\mathbf{Q}}\hookrightarrow \mathbf{C}$ and $\overline{\mathbf{Q}}\hookrightarrow \overline{\mathbf{Q}}_p$ on which these isomorphisms depend? If you then change one but not the other, wouldn't your composition change?

Bosch’s book “Lectures on Formal and Rigid Geometry” is a good start. And Kedlaya’s lectures in the AWS volume on perfectoid spaces.

Even worse, $F$ does not a priori act on the cohomology of some sheaf $\mathcal{F}$ (unless e.g. $\mathcal{F}$ is constant). Rather, it induces a map $F^* \colon H^*(X, \mathcal{F}) \to H^*(X, F^* \mathcal{F})$ between two different groups. To compare it with the identity, you need an "equivariant structure" i.e. some choice of isomorphism $F^*\mathcal{F}\to \mathcal{F}$. In case of the Frobenius, such a canonical structure is described in the paragraph preceding the result in SGA5 you cite. P.S. $F$ is affine since the preimage of an affine open $U$ is $U$, which is affine.

I disagree with the statement in the second paragraph: consider the case when $F$ is an isomorphism. Certainly not every automorphism induces the identity on cohomology! Did you mean to assume that $F$ is the identity on the underlying space? But then, the case of the Zariski topology is trivial.

By flatness of $Y\to X$, a regular sequence in a local ring on $X$ produces a regular sequence in a local ring on $Y$.

Fppf-crystalline could be tricky as typically you use smoothness a lot, and smoothness is not fppf local

I imagine you already checked in Olsson’s Asterisque on crystalline cohomology of stacks?

Is $X$ assumed to be regular (at least at every point of $D$)?

A possible source of lots of such sheaves: if $X$ is over $\mathbf{F}_p$ and $F\colon X\to X$ is the Frobenius, then for any given $n\geq 0$ there is an $e\geq 0$ such that for every coherent sheaf $\mathcal{F}$ on $X$, the $e$-th pullback $(F^e)^* \mathcal{F}$ has this property.