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Many recent studies of Entity Alignment (EA) use Graph Neural Networks (GNNs) to aggregate the neighborhood features of entities and achieve better performance. However, aligned entities in real Knowledge Graphs (KGs) usually have non-isomorphic neighborhood structures due to the different data sources of KGs. Therefore, it is insufficient to simply compare the global direct neighborhood of aligned entities, which may also become a variable for the EA judgment. In this paper, we propose a Relation-based Adaptive Neighborhood Matching method (RANM), which matches larger range and higher confidence neighborhoods for aligned entities based on relation matching instead of alignment seeds. RANM first uses alignment seeds to construct the best relation matching set, and then performs local direct neighborhood matching and feature aggregation on the candidate alignments. To obtain high-quality entity embeddings, we design a variant attention mechanism based on heterogeneous graphs, which considers the heterogeneity of relations in KGs. We also adopt a bi-directional iterative co-training to further improve the performance. Extensive experiments on three well-known datasets show our method significantly outperforms 14 state-of-the-art methods, and is 3.01-11.5% higher than the best-performing baselines in Hits@1. RANM also shows high performance on the long-tailed entities and the dataset with less alignment seeds.