Jacobian fibration of an abelian fibration

Let $$f \colon S \rightarrow C$$ be a minimal elliptic surface and let $$g \colon J \rightarrow C$$ be its jacobian fibration. In this case, we know that the fibers of $$g$$ are better behaved that the ones of $$f$$. There is an explicit classification of what fiber one gets after taking the jacobian fibration (see, for instance, the chapter on elliptic surfaces in Algebraic Surfaces, by Shafarevich, Proceedings of the Steklov Institute of Mathematics. American Mathematical Society, 75). This information can be rendered numerically by means of the log canonical threshold: for every $$o \in C$$, $$\mathrm{lct}(J,g^*o) \geq \mathrm{lct}(S,f^*o)$$ (see Lemma 1.6 here for the inequality between boundary divisors in the canonical bundle formula, https://arxiv.org/pdf/alg-geom/9305002.pdf, and section 3 here for the relation between lct and the coefficients of the boundary divisors https://arxiv.org/pdf/1210.5052.pdf).

My question is the following. Do we have a similar description for fibrations of higher-dimensional abelian varieties? We can assume we are in the following situation: C is a smooth curve, $$f\colon X\rightarrow C$$ is a relatively minimal fibration whose geometric generic fiber is an abelian variety, and $$g\colon J\rightarrow C$$ is a relatively minimal model of the corresponding jacobian fibration.

1) Do we still have the inequality between the log canonical thresholds?

2) Can we get an explicit description of what the new fiber is (maybe just in some cases)?