My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).

In such a setup, one has a fibration $f: X \rightarrow Y$, and a log canonical pair $(X,B)$, which is $\mathbb{Q}$-linearly trivial over $Y$ (i.e. $K_X+B \sim_{\mathbb{Q},f} 0$). One can construct naturally induced divisors on $Y$: a boundary divisor $B_Y$ and a pseudoeffective divisor $M_Y$. Also, one can induce divisors $B_{Y'}$ and $M_{Y'}$ for any higher model of $Y$.

If the coefficients of $B$ are rational (which is implicit by the $\mathbb{Q}$-linear equivalence above), one has that $M_{Y'}$ is nef on a high enough model, and the further higher $M_{Y''}$ are obtained by pullback.

Thus, I am looking for some concrete example of the above behavior. I went through the literature, but I could not find explicit constructions where one needs to go to a higher model to realize nefness. Could somebody provide or refer to any?

Also, is it possible that $M_Y$ is already nef, yet the higher $M_{Y'}$'s are not its pullback?

Also, more generally, I am interested in concrete examples of birational morphisms where the pushforward of a nef divisor is no longer nef.