  We study the integrability of Poisson and Dirac structures that arise from
quotient constructions. From our results we deduce several classical results as
well as new applications. We also give explicit constructions of Lie groupoids
integrating two interesting families of geometric structures: (i) a special
class of Poisson homogeneous spaces of symplectic groupoids integrating Poisson
groups and (ii) Dirac homogeneous spaces.
