Let $X$ and $Y$ be irreducible projective varieties. I would like to show that if $f: X \rightarrow Y$ is generically finite and $M \rightarrow Y$ is an ample line bundle, then $L = f^{\ast} M$ is big. Morally I think this is because $f$ is "finite on an open dense set" and so $L$ is "ample on an open dense set" and therefore some power gives an embedding that must have the right dimension.
But I can't quite complete the argument, first because I guess this is not precisely what generically finite means, and second since in order to say that a pullback of ample by finite is ample, I need surjectivity of $f$ or compactness of both spaces (Corollaries 1.2.13 and 1.2.28 in Lazarsfeld), both of which break when I pass to an open subset.
For reference this statement is made right under Definition 2.2.1 in Lazarsfeld.