Liftable derivations for generically separably algebraic morphisms of schemes

We consider dominant, generically algebraic, and tamely ramified (if the characteristic is positive) morphisms $\pi: X/S \to Y/S$, where Y,S are Noetherian and integral and X is a Krull scheme (e.g. normal Noetherian), and study the sheaf of tangent vector fields on Y that lift to tangent vector fields on X. We give an easily computable description of these vector fields using valuations along the critical locus. We generalise to positive characteristic Seidenberg's theorem on lifting tangent vector fields to the normalisation. If $\pi$ is a blow-up of a coherent ideal I we show that tangent vector fields that preserve the Ratliff-Rush hull of I are liftable, and that all liftable tangent vector fields preserve the integral closure of I.