  We reexamine the notions of generalized Ricci tensor and scalar curvature on
a general Courant algebroid, reformulate them using objects natural w.r.t.
pull-backs and reductions, and obtain them from the variation of a natural
action functional. This allows us to prove, in a very general setup, the
compatibility of the Poisson-Lie T-duality with the renormalization group flow
and with string background equations. We thus extend the known results to a
much wider class of dualities, including the cases with gauging (so called
dressing cosets, or equivariant Poisson-Lie T-duality). As an illustration, we
use the formalism to provide new classes of solutions of modified supergravity
equations on symmetric spaces.
