  Let $R$ be a finitely generated positively graded algebra over a Noetherian
local ring $B$, and $\mathfrak{m} = [R]_+$ be the graded irrelevant ideal of
$R$. We provide a local criterion characterizing the $B$-freeness of all the
local cohomology modules $\text{H}_\mathfrak{m}^i(M)$ of a finitely generated
graded $R$-module $M$. We show that fiber-full modules are exactly the ones
that satisfy this criterion. When we change $B$ by an arbitrary Noetherian ring
$A$, we study the fiber-full locus of a module in $\text{Spec}(A)$: we show
that the fiber-full locus is always an open subset of $\text{Spec}(A)$ and that
it is dense when $A$ is generically reduced.
