$\def\sF{\mathcal{F}} \def\sO{\mathcal{O}} \def\spec{\operatorname{Spec}} \def\frp{\mathfrak{p}} \def\im{\operatorname{Im}}$Consider the following result:
Lemma. Let $\sF$ be a quasi-coherent sheaf of modules over a scheme $X$, and for a point $x\in X$, denote $j:\spec\sO_{X,x}\to X$ to the canonical morphism. Suppose that $X$ is affine or that $x$ is the generic point of an irreducible component. Then we have an isomorphism $$ \sF\otimes_{\sO_X}j_*\sO_{\spec\sO_{X,x}}\cong j_*\widetilde{\sF_x} $$ of $j_*\sO_{\spec\sO_{X,x}}$-modules.
My sole question is:
Is the lemma still true if $X$ is non-affine and $\overline{\{x\}}$ is not an irreducible component? (or at least, is there at least more general sufficient conditions than the ones above for the lemma to hold?)
I guess to look for a counterexample one should answer first what's $j_*\sO_{\spec\sO_{X,x}}$ when $\overline{\{x\}}$ is not an irreducible component, but I don't see a pattern after looking to some examples.
For completeness, I'll provide:
Proof of the lemma. If $X=\spec A$ is affine then $\sF\cong\widetilde{M}$ for some $A$-module $M$. Denote $\frp\subset A$ to the prime associated to $x$. The isomorphism of the statement is the $A$-tildification of the canonical isomorphism $M\otimes_AA_\frp\cong M_\frp$ (use 01I8(1) and 01I9(2)).
Now suppose $X$ is any scheme and $x=\eta$ is the generic point of an irreducible component $T=\overline{\{\eta\}}$ of $X$. We are going to construct a canonical morphism $\varphi:\sF\to j_*\widetilde{\sF_x}$ of $\sO_X$-modules. Suppose $U\subset X$ is an affine open neighborhood of $x$. Define $\varphi_U$ to be the $A$-tildification of the canonical $A$-linear map $M\to M_\frp$ (use 01I9(2)). On the other hand, note that $\im j=\{\eta\}$. Hence, we define $\varphi_{X\setminus T}=(\sF|_{X\setminus T}\to 0)$, for $j_*\widetilde{\sF_x}|_{X\setminus T}=0$. Finally, if $y\in T$, then any neighborhood of $y$ contains $\eta$. In particular, $X\setminus T$ plus the open affine neighborhoods of $\eta$ cover $X$.
These morphisms glue: suppose $U,V\subset X$ are open affine neighborhoods of $x$. We want to see $\varphi_U|_{U\cap V}=\varphi_V|_{U\cap V}$. By Nike's trick, it suffices to show it for $V\subset U$ and $V$ distinguished affine in $U$. This follows from the fact that the canonical morphism $M\to M_\frp$ commutes with localizations. On the other hand, if $U\subset X$ is an open affine neighborhood of $\eta$, then $\varphi_U$ and $\varphi_{X\setminus T}$ agree on $U\cap(X\setminus T)$, for $j_*\widetilde{\sF_x}|_{X\setminus T}=0$.
By the restriction-extension of scalars adjunction relative to $\sO_X\to j_*\sO_{\spec\sO_{X,x}}$, the adjunct of $\varphi$ is a morphism $$ \psi:\sF\otimes_{\sO_X}j_*\sO_{\spec\sO_{X,x}}\to j_*\widetilde{\sF_x} $$ of $j_*\sO_{\spec\sO_{X,x}}$-modules. On $X\setminus T$, this morphism restricts to $0\to 0$, since $j_*\mathcal{G}|_{X\setminus T}=0$ for any module $\mathcal{G}$ over $\spec\mathcal{O}_{X,x}$. On an open affine neighborhood $U$ of $x$, we have that $\psi|_U$ is the map from the very first paragraph of the proof. $\square$