  We establish a 1:1 correspondence between Poisson-Lie group actions on
integrable Poisson manifolds and twisted multiplicative hamiltonian actions on
source 1-connected symplectic groupoids. For an action of a Poisson-Lie group
$G$ on a Poisson manifold $M$, we find an explicit description of the lifted
hamiltonian action on the symplectic groupoid $\Sigma(M)$. We give applications
of these results to the integration of Poisson quotients $M/G$, Lu-Weinstein
quotients $\mu^{-1}(e)/G$ and Poisson homogeneous spaces $G/H$.
