Here are two theorems:
If $F$ is a finite type quasi-coherent sheaf on a scheme $X$, then $ \operatorname{rank}(F)(p) = \dim_{k(p)} F_p \otimes_{O_p} k(p)$ is a upper semicontinuous function on $X$.
If $\pi : X \to Y$ is a closed morphism of finite type $k$ schemes, then the dimension of the fiber over $q \in Y$ is an upper semicontinuous function on $Y$. (Also, the dimension of the largest component of the fiber containing $P \in X$ is an upper semicontinuous function on X.)
Are these related?
For a finite type sheaf $F$ on a reduced scheme X, it looks like a vector bundle on each of the locally closed subsets given by $\operatorname{rank} F = k$ for $k$ varying. So in this case my guess would be that there is a total space over $X$ for which $F$ is the sheaf of sections and the two theorems — in the case of a finite type reduced $k$-scheme — are saying the same thing here.
When is there a reasonable total space for a quasicoherent sheaf? (The étale space is not reasonable — it is way to big.)
Are they both special cases of something more general?
Why does upper semi-continuity of fibers appear so often?