I have some questions related to base change in cohomology of schemes and formal schemes, in particular related to base change along fibers. Sorry if this question is too easy for this forum, but I cannot find any reference answering explicitly to my question. Let me explain the questions in details. First, consider a map $f:X\rightarrow Y$ of schemes, with $Y=\text{Spec}(A)$ affine, and let $\mathcal{F}$ be a coherent sheaf over $X$. Let $x:\text{Spec}(k)\rightarrow Y$ be a point (maybe not a geometric point) of $Y$ and consider the base change $X_k$ along the morphism $X$. Call $g:X_k\rightarrow X$ the map induced by base change. Under which conditions on all this setting, is it possible to conclude that $H^i(X,\mathcal{F})\otimes_A k\cong H^i(X_k,g^\ast\mathcal{F})$? In particular, the situation I'm interested in is the case when $Y=\text{Spec}(\mathbb{Z}_p)$, and $x$ is the special fiber, so I'm trying to exclude condition of flatness for the morphism $x$. I'm also interested in some spectral sequences giving an approximation to this situation.
The next question, related to the first one, concerns with formal schemes. Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of formal schemes. Assume that $\mathcal{Y}$ comes as the completion of an affine scheme along a closed subscheme, say $\mathcal{Y}$ is the completion of $Y=\text{Spec}(A)$ along the susbscheme defined by the ideal $\mathcal{I}=\tilde{I}$, for $I$ the ideal of definition of $A$. Again, are there any suitable condition, or maybe a spectral sequence, under which $H^i(\mathcal{X}, \mathcal{F})\otimes_{A}A/I\cong H^i(\mathcal{X}\times_{\text{Spf}A}\text{Spf}(A/I),g^\ast(\mathcal{F}))$, where, as before, $\mathcal{F}$ is a quasi coherent sheaf of $\mathcal{O}_{\mathcal{X}}$-modules and $g:\mathcal{X}\times_{\text{Spf}(A)}\text{Spf}(A/I)\rightarrow \mathcal{X}$ is the map induced by pullback? Of course I assume that $A/I$ has the discrete topology. Again, the situation I'm mainly interested in is the one where $\mathcal{Y}=\text{Spf}(\mathbb{Z}_p)$ and $\mathcal{X}$ is a maybe not algebraizable $p$-adic formal scheme, and $I$ is the ideal $p\mathbb{Z}_p$.
Does these conclusions hold at least for $i=0$?