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Let $f:Y\to X$ be a finite morphism between Fano manifolds $Y$ and $X$ suchthat 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 $f$ isbounded 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 allthe singular quadrics admitting non-isomorphic endomorphisms.

Boundedness of finite morphisms onto Fano manifolds with large Fano index