Let $k$ be an algebraically closed field of characteristic zero. We consider $f:\mathfrak{X}\to \mathbb{A}^1$ an smooth projective morphism of pure relative dimension $n$, where $\mathfrak{X}$ is an smooth algebraic variety.

Suppose that $\mathcal{F}$ is a torsion-free coherent sheaf on $\mathfrak{X}$ which is flat over $\mathbb{A}^1$. Then, $f_*\mathcal{F}\cong \text{H}^0(\mathfrak{X},\mathcal{F})^{\sim}$ is a torsion-free coherent sheaf on the smooth curve $\mathbb{A}^1$, which is therefore locally free. Even better, algebraic vector bundles over the affine line are trivial.

**My question is the following**: Suppose that $\text{H}^0(\mathfrak{X}_t,\mathcal{F}_t)$ are all isomorphic and that the map $f_*\mathcal{F}\otimes \kappa(t) \to \text{H}^0(\mathfrak{X}_t,\mathcal{F}_t)$ is an isomorphism, for every closed point $t\neq 0$.

What can we say about $h^0(\mathfrak{X}_0,\mathcal{F}_0)$ in this particular situation or, equivalently (by Mumford "Abelian varieties" II.5 Corollary 2), can we expect $f_*\mathcal{F}\otimes \kappa(0)\to \text{H}^0(\mathfrak{X}_0,\mathcal{F}_0)$ to be an isomorphism ? Maybe an injection ("by semi-continuity") ? Under additional hypotheses ?

Thank you in advance for any comment.