Liftable derivations for generically separably algebraic morphisms of schemes

We consider dominant, generically algebraic (e.g. generically finite), and tamely ramified (if the characteristic is positive) morphisms $\pi : X/S \to Y/S$ of $S$-schemes, where $Y,S$ are Nœtherian and integral and $X$ is a Krull scheme (e.g. normal Nœtherian), 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 apply this to answer the question when the liftable derivations can be defined by a tangency condition along the discriminant. In particular, if $\pi$ is a blow-up of a coherent ideal $I$, we show that tangent vector fields that preserve the Ratliff-Rush ideal (equals $[I^{n+1}:I^n]$ for high $n$) associated to $I$ are liftable, and that all liftable tangent vector fields preserve the integral closure of $I$. We also generalise in positive characteristic Seidenberg's theorem that all tangent vector fields can be lifted to the normalisation, assuming tame ramification.