  We prove the universal lifting theorem: for an $\alpha$-simply connected and
$\alpha$-connected Lie groupoid $\gm$ with Lie algebroid $A$, the graded Lie
algebra of multi-differentials on $A$ is isomorphic to that of multiplicative
multi-vector fields on $\gm$. As a consequence, we obtain the integration
theorem for a quasi-Lie bialgebroid, which generalizes various integration
theorems in the literature in special cases.
  The second goal of the paper is the study of basic properties of
quasi-Poisson groupoids. In particular, we prove that a group pair $(D, G)$
associated to a Manin quasi-triple $(\mathfrak d, \mathfrak g, \mathfrak h)$
induces a quasi-Poisson groupoid on the transformation groupoid $G\times
D/G\toto D/G$. Its momentum map corresponds exactly with the $D/G$-momentum map
of Alekseev and Kosmann-Schwarzbach.
