Let $X$ and $Y$ be schemes of finite type over $\mathbb{C}$. Is it true that if a morphism $f:X\rightarrow Y$ is injective, then $f$ is universally injective? (I recall having read this somewhere, but cannot find the reference.) If not, then under what conditions will injective imply universally injective?

(If you like, you may take $X$ and $Y$ to be smooth quasi-affine varieties, and $f:X\rightarrow Y$ to be a smooth morphism.)