Excuse me if this is a naive question. Let $f : X \to Y$ be a morphism of varieties over a field $k$ and $\mathcal{F}$ a quasi-coherent sheaf on $X$. I know that for general sheaves on spaces not much can be said about the stalk $(f_*\mathcal{F})_y$ at $y \in Y$ (let's just talk about closed points), but does anything nice happen in this situation? If $f$ is proper then the completion of the stalk is described by the formal function theorem, but I'm interested in the honest stalk.
Suppose, for instance, that $X = \text{Spec } A$ is affine (so $\mathcal{F} = \widetilde{M}$ for some $A$-module $M$) and that the scheme-theoretic fiber of $f$ over the closed point $y \in Y$ is reduced. Write this fiber as a union of irreducible components $Z_1 \cup \cdots \cup Z_r$ corresponding to prime ideals $\mathfrak{p}_1,\cdots,\mathfrak{p}_r \subset A$, so there is an associated semi-local ring $S^{-1}A$ with $S = A \setminus (\mathfrak{p}_1 \cup \cdots \cup \mathfrak{p}_r)$. Hopefully here $(f_*\mathcal{F})_y$ is just $S^{-1}M$ regarded as an $\mathcal{O}_{Y,y}$-module via the natural map $\mathcal{O}_{Y,y} \to S^{-1}A$.
Does this make sense? Is something like this true when $X$ is not affine and/or the fiber is nonreduced?
Edit: As Georges's answer shows, this cannot possibly be true in general. I wonder if there is still hope when $f$ is proper? My example in the comments below (the squaring map $\mathbb{A}^1 \to \mathbb{A}^1$ with $\mathcal{F} = \mathcal{O}$ and $y = 1$) is actually consistent with my guess above: $(f_*\mathcal{F})_y$ is the localization of $k[t]$ at $k[t^2] \setminus (t^2-1)k[t^2]$. It is not hard to see that this coincides with the localization at $k[t] \setminus ((t-1) \cup (t+1))$, i.e. the semilocal ring at $\{ \pm 1 \}$.