Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties, and $NE(X),NE(Y)$ the Mori cones of curves of $X$ and $Y$. Assume that $NE(X)$ is finitely generated. Then is $NE(Y)$ finitely generated as well?

If $f$ is birational I think the answer is positive. Let $C\subset Y$ be an irreducible curve and $\Gamma$ its strict transform in $X$. Then $\Gamma\sim a_1\Gamma_1+\dots + a_r\Gamma_r$ with $a_1,...,a_r\geq 0$ and where $\Gamma_1,\dots,\Gamma_r$ are the generators of $NE(X)$. So $C\sim a_1f_{*}\Gamma_1 + \dots + a_rf_{*}\Gamma_r$ and hence $NE(Y)$ is generated by $f_{*}\Gamma_1,\dots,f_{*}\Gamma_r$. Is this argument correct?

If $f$ is not birational could it happen that $NE(Y)$ is not finitely generated even if $NE(X)$ is?

Thank you very much.