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