Let $f:X\rightarrow Y$ be a projective morphism between normal quasi-projective varieties, where $X$ is $\mathbb{Q}$-factorial. A Cartier divisor $D$ on $X$ is called $f$-big if $D|_{X_\eta}$ is big, where $X_\eta$ is the fibre of the generic point of $Y$.
My question is: If in addition the above $f$ is birational, is every divisor on $X$ automatically $f$-big? If so, why?
BTW, The question arised when I was reading the proof of the existence of flips in [BCHM].