Vanishing of Higher Direct Images Let $X$ and $Y \ $ be smooth varieties over a field or - depending on the answers - more general nice schemes (I don't know what one needs exactly as conditions).
Let $p: X\times Y \rightarrow X$ be the projection morphism. It has constant fiber dimension $n:=dim(Y)$ over$X$, so is smooth of relative dimension n.
Is it then right that one has a vanishing of the higher direct images for any quasicoherent sheaf $\mathcal F$ on $X\times Y$:
$\mathcal R^i \mathcal F =0$ for all $i > n$ ?
Note that by Grothendieck's Vanishing Theorem one surely has it for $i>n+dim(X)$.
Also I really want Quasicoherence, not Coherence, so Semicontinuity Arguments don't seem to be at hand.
Furthermore, how far can one generalize this?
 A: I am not sure what you call semicontinuity theorems. Your question is about vanishing of $R^i$'s. 

Let $f : X\to S$ be a proper morphism to a locally noetherian scheme $S$ such that all its fibers have dimension $\le n$. Let $F$ be a quasi-coherent sheaf on $X$. Then $R^if_*F=0$ for all $i> n$. 

Proof (admitting the result for coherent sheaves on $X$). The question is local on $S$, so we can suppose $S$ is affine. Then we have to show that $H^i(X, F)=0$ for $i>n$. Fix a finite affine covering $\mathcal U$ of $X$.  We will compute $H^i(X,F)$ by Cech cohomology $H^i(\mathcal U, F)$. Let 
$$c\in \mathrm{ker}(C^{i}(\mathcal U, F)\to C^{i+1}(\mathcal U, F)).$$ 
By EGA I.9.4.9, $F$ is the inductive limit of its coherent submodules $(F_{\alpha})_{\alpha}$. As $c$ involves finitely many sections, there exists $\alpha$ such that 
$$c\in\mathrm{ker}(C^{i}(\mathcal U, F_\alpha)\to C^{i+1}(\mathcal U, F_\alpha))=\mathrm{Im}(C^{i-1}(\mathcal U, F_\alpha)\to C^{i}(\mathcal U, F_\alpha)).$$
Hence 
$$c \in \mathrm{Im}(C^{i-1}(\mathcal U, F)\to C^{i}(\mathcal U, F))$$
and $H^i(X,F)=H^i(\mathcal U, F)=0$.  
Remark For a direct proof (without passing by coherent sheaves) in the case of projective morphisms to a locally noetherian scheme (I was wrong on quasi-projective morphisms), see Proposition 5.2.34 in "Algebraic geometry and arithmetic curves". 
