This paper addresses the problem of matching common node correspondences among multiple graphs referring to an identical or related structure. This multi-graph matching problem involves two correlated components: i) the local pairwise matching affinity across pairs of graphs; ii) the global matching consistency that measures the uniqueness of the pairwise matchings by different composition orders. Previous studies typically either enforce the matching consistency constraints in the beginning of an iterative optimization, which may propagate matching error both over iterations and across graph pairs; or separate affinity optimization and consistency enforcement into two steps. This paper is motivated by the observation that matching consistency can serve as a regularizer in the affinity objective function especially when the function is biased due to noises or inappropriate modeling. We propose composition-based multi-graph matching methods to incorporate the two aspects by optimizing the affinity score, meanwhile gradually infusing the consistency. We also propose two mechanisms to elicit the common inliers against outliers. Compelling results on synthetic and real images show the competency of our algorithms.