Take the bundle $\mathcal{O}{\mathbb{P}^1} \otimes \mathcal{O}{\mathbb{P}^2}(3)$ on $\mathbb{P}^1 \times \mathbb{P}^2$. The generic section is a K3, call it $M$, and let $p : \mathbb{P}^1 \times \mathbb{P}^2 \to \mathbb{P}^1$ be the projection onto the first factor. The restriction of $p$ to $M$ gives an elliptic fibration with generic fiber an elliptic curve, total space a $K3$ and base $\mathbb{P}^1$.

From this, we can also produce an example of a Calabi—Yau fiber space with non-maximal variation and which is not isotrivial. The point is to consider a threefold $X$ of Kodaira dimension $2$ whose Iitaka map is not isotrivial. Indeed, in this case, the Iitaka map $\Phi : X \to B$ is an elliptic fiber space with $\dim(B) =2$ and generic fiber and elliptic curve. Let $\mu : B \to \mathcal{M}$ be the moduli map from the base of the family into the moduli space of elliptic curves. The differential $d\mu : \mathcal{T}_B \to \mathcal{T}M$ is the Kodaira—Spencer map which cannot be injective for dimension reasons: $\dim {\mathcal{T}{\mathcal{M}}} =1 < 2 = \dim \mathcal{T}_B$.

To construct a threefold of Kodaira dimension $2$, apply a logarithmic transformation to the K3 surface $M$ given above. Blow it up $k>0$ times to produce a surface $\widetilde{M}$. Take the cartesian product with a curve $C$ of genus $g \geq 2$. Then $\widetilde{M} \times C$ is a threefold with Kodaira dimension $2$.

Acknowledgements: This approach to constructing examples of Calabi—Yau fiber spaces with intermediate variation was told to me by Frédéric Campana and Behrouz Taji. The construction was also inspired by conversations with Yanir Rubinstein.