We globalize the derived version of the McKay correspondence of
Bridgeland-King-Reid, proven by Kawamata in the case of abelian quotient
singularities, to certain log algebraic stacks with locally free log structure.
The two sides of the correspondence are given respectively by the infinite root
stack and by a certain version of the valuativization (the projective limit of
every possible log blow-up). Our results imply, in particular, that in good
cases the category of coherent parabolic sheaves with rational weights is
invariant under log blow-up, up to Morita equivalence.