Divisor question on the normal projective curve

Question

Let $X$ a normal projective curve over an infinite field $k$, let $x_1,\dots,x_n$ be pairwise distinct closed points in $X$ and let $n_1,\dots,n_r\in\mathbb Z$. Let $$D=\sum_in_ix_i$$ (considered as a Cariter divisor).

Let $y\in X$ be a closed point distinct from all the $x_i$. Show that for large enough $n$, we have $$L(D+ny) \ne \bigcup_iL(D+ny-x_i).$$

It is the Ex7.3.2 a) from the book Algebraic Geometry and Arithmetic Curves by Qing Liu. I add the following qusetions.

1. If we treat $D$ as a Weil divisor, what is the difference?
2. It seems that if $n$ is not large enough, we may have $$L(D+ny) =\cup_iL(D+ny-x_i),$$ some gap phenomenon？How to explian it?