I think Corollary 4.3 of Spreafico's [Axiomatic theory for transversality and Bertini type theorems](http://www.springerlink.com/content/8fq21fpdnv030h9w/) does what you want. It says (in the case where the property is taken to be smoothness) that if $f:X\to \mathbb P^n$ is a finite type morphism from a smooth scheme $X$ over any infinite field, and if $f$ is *residually separated* (i.e. the induced extensions of residue fields are separable), then the pullback of a generic hyperplane is smooth.