  In this paper we investigate the degrees of irrationality of degenerations of
$\epsilon$-lc Fano varieties of arbitrary dimensions. We show that given a
generically $\epsilon$-lc klt Fano fibration $X\to Z$ of dimension $d$ over a
smooth curve $Z$ such that $(X, t F)$ is lc for a positive real number $t$
where $F$ is the reduction of an irreducible central fibre of $X$ over a closed
point $z\in Z$, then $F$ admits a rational dominant map $\pi\colon
F\dashrightarrow C$ to a smooth projective variety $C$ with bounded degree of
irrationality depending only on $\epsilon, d, t$ such that the general fibres
of $\pi$ are irreducible and rational. This proves the generically bounded case
of a conjecture proposed by the first author and Loginov for log Fano
fibrations of dimensions greater than three. One of the key ingredients in our
proof is to modify the generically $\epsilon$-lc klt Fano fibration $X\to Z$ to
a toroidal morphism of toroidal embeddings with bounded general fibres.
