  Let $f:Y\to X$ be a finite morphism between Fano manifolds $Y$ and $X$ such
that the Fano index of $X$ is greater than 1. On the one hand, when both $X$
and $Y$ are fourfolds of Picard number 1, we show that the degree of $f$ is
bounded in terms of $X$ and $Y$ unless $X\cong\mathbb{P}^4$; hence, such $X$
does not admit any non-isomorphic surjective endomorphism. On the other hand,
when $X=Y$ is either a fourfold or a del Pezzo manifold, we prove that, if $f$
is an int-amplified endomorphism, then $X$ is toric. Moreover, we classify all
the singular quadrics admitting non-isomorphic endomorphisms.
