We prove the universal lifting theorem: for an $\alpha$-simply connected and $\alpha$-connected Lie groupoid $\Gamma$ with Lie algebroid $A$, the graded Lie algebra of multi-differentials on $A$ is isomorphic to that of multiplicative multi-vector fields on $\Gamma$. 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,...