Inverse image of a $0$-dimensional subscheme under a finite morphism

Question

Let $f\colon X\rightarrow Y$ be a finite surjective morphism of complex projective varieties of the same dimension. We may suppose $Y$ is smooth. Let $Z\subset Y$ be a $0$-dimensional closed subscheme and let $Z'=X\times_Y Z$ be the scheme-theoretic inverse image of $Z$ under $f.$ Since $f$ is finite, $Z'\subset X$ is closed of dimension $0.$ If $\mathrm{lt}(Z):=h^0(Z,\mathcal{O}_Z)$ is the length of $Z,$ is it true that $\mathrm{lt}(Z)\leq \mathrm{lt}(Z')$?