  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.
