Let $f:X\rightarrow \mathbb{P}_1$ be a smooth fibration from a smooth rationally connected manifolds to the smooth rational curve $\mathbb{P}_1$. Assume further that the generic fiber of $f$ is also rationally connected.
Question: Besides the trivial product of $\mathbb{P}_1$ with a rationally connected manifolds, is there some examples of such fibration which is not locally isotrivial? Thanks!
I am just posting my comment as an answer.
The simplest examples are obtained as follows. Begin with $Y=\mathbb{P}^2\times \mathbb{P}^1$ together with its projection $$\text{pr}_2: \mathbb{P}^2\times \mathbb{P}^1 \to \mathbb{P}^1.$$ For each integer $N\geq 5$, let $(\sigma_i:\mathbb{P}^1\to \mathbb{P}^2\times \mathbb{P}^1)_{i=1,\dots,N}$ be a collection of pairwise disjoint sections of $\text{pr}_2$ such that for a general pair $(a,b)\in \mathbb{P}^1\times\mathbb{P}^1$, $(\mathbb{P}^2,\{\sigma_1(a),\dots,\sigma_N(a)\})$ is not projectively equivalent to $(\mathbb{P}^2,\{\sigma_1(b),\dots,\sigma_N(b)\})$. Let $\nu:X\to Y$ be the blowing up of $X$ along the ideal sheaf of the union of the sections $\sigma_i$. Let $f$ be $\text{pr}_2\circ \nu$.
In particular, if $N=5,6,7,8$ and if the sections are sufficiently general, then every fiber of $\nu$ has nef, big, and globally generated anticanonical bundle. Moreover, the general fiber is Fano.
If you want examples where every fiber is Fano, these do also exist, but the examples that I know are isotrivial over a dense open subset of the base. The idea is to consider specializations of projective homogeneous spaces of Picard rank one to Fano manifolds of Picard rank one that are not homogeneous. Such examples were discovered by Pasquier-Perrin.
