For each $i\in I$, let ${\mathscr{X}}_{i}$ be the closed subscheme of ${\mathscr{X}}$ defined as the topological closure of $X_{i}$ in ${\mathscr{X}}$ equipped with the reduced structure. We then get for each $i\in I$ a cartesian diagram

Since the morphism $\coprod_{i\in I}{\mathscr{X}}_{i}\to{\mathscr{X}}$ is finite surjective, the projection formula shows that ${\mathscr{L}}$ is nef on ${\mathscr{X}}$ if and only if ${\mathscr{L}}_{|{\mathscr{X}}_{i}}$ is nef on ${\mathscr{X}}_{i}$ for all $i$. ∎