I think Corollary 4.3 of Spreafico's [Axiomatic theory for transversality and Bertini type theorems][1] 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.

  [1]: https://doi.org/10.1007/s000130050213 "Arch. Math. 70, 407–424 (1998). zbMATH review at https://zbmath.org/0936.14013"