Let $f:X\to S$ be a smooth proper morphism of schemes with geometrically connected fibres. Assume $S$ is a smooth irreducible variety over $\mathbb{C}$. Assume that there is a sequence of closed points $(s_i)_{i=1}^\infty$ in $S$ such that the fibres $X_{s_i}$ are pairwise non-isomorphic over $\mathbb{C}$.

Is there a curve $C\subset S$ such that the restricted family $f|_{X_C}:X_C\to C$ is non-isotrivial? (That is, are there infinitely many points $c_i$ such that the fibres $X_{c_i}$ are pairwise non-isomorphic?)

I think the answer is yes, and that one can construct $C$ by taking a "general" curve. But how does one make this precise?