Here is a counterexample (vaguely inspired by Kevin Casto's approach, and very similar to nfdc23's example). The moduli space $\mathcal M_g$ of smooth genus $g \geq 2$ curves is irreducible, and has a closed subset corresponding to curves admitting a nontrivial automorphism (this is a strict subset if $g \geq 3$). Thus we can form a family whose general member has no automorphisms, but a special member is, say, hyperelliptic.

That is, we can produce a family $\mathscr C \to \operatorname{Spec} \mathbb C[x]_{(x)}$ whose generic fibre $C$ has no automorphisms and whose special fibre $C_0$ has a hyperelliptic involution $\sigma \colon C_0 \to C_0$ that does not lift to an automorphism of $C$. Let $p \in C_0$ be a point not fixed by $\sigma$, and consider the flat (but non-proper) families $\mathscr C \setminus\{p\}$ and $\mathscr C\setminus\{\sigma(p)\}$ over $\operatorname{Spec}\mathbb C[x]_{(x)}$.

Now the generic fibres are both $C$, and the special fibres are $C_0 \setminus\{p\}$ and $C_0\setminus\{\sigma(p)\}$, which are isomorphic through $\sigma$. But a global isomorphism would induce an automorphism of $C$, which therefore has to be the identity on $C$. The only extension of the identity to a map $\mathscr C \setminus \{p\} \to \mathscr C$ is the inclusion, which does not land in $\mathscr C \setminus\{\sigma(p)\}$. (Informally: '$p$ cannot go to $\sigma(p)$'.)

**Remark.** I do not have an example of a proper family. In fact, all the counterexamples suggested so far are non-proper.