API migration is essential for software maintenance due to the rapid evolution of third-party libraries where API elements may change continuously through updates. There are two main challenges for API migration at the project level, especially across multiple versions: 1) lack of specific library evolution knowledge across multi-version; 2) difficulty in identifying the chain of changes at the project level. This paper proposes a project-level cross-multi-version API migration framework APIMig. We first construct an API evolution knowledge graph (KG) to capture changes between adjacent library versions and then derive coherent cross-version API evolution knowledge by KG reasoning. Second, we design a chain exploration algorithm to track the chain of changes and aggregate the affected code segments. Finally, a large language model is employed in completing API migration by providing the API evolution knowledge and the chain of changes. We construct an evolution KG for the Lucene library from version 4.0.0 to 10.1.0 and evaluate our approach through project migration pairs that depend on different major versions. Our framework shows improvements over the baseline in migrating projects across 7 major versions, achieving average increases of 16.52% in CodeBLEU scores and 28.49% in VCEU scores in GPT-4o.