Let $X$ be an integral Noetherian scheme that has no automorphisms other than the identity. Is it possible to characterize (in a not completely tautological way) $X$ such that

- there is no finite morphism $X\rightarrow X$ surjective on the underlying topological spaces (other than the identity)?
- there is no finite flat morphism $X\rightarrow X$ surjective on the underlying topological spaces (other than the identity)?

If $X$ is regular, then there is no distinction between two bullet points.

Among spectra of fields, one can find examples satisfying the first (eq. second) bullet point ($\mathrm{Spec}\,\mathbb{Q}$) or not satisfying the first (eq. second) bullet point ($\mathrm{Spec}\,k$ for $k$ rigid non-perfect).