This is the expected behavior since the Document ID links are redirect links for the documents.
Document IDs is meant for providing an easy way to track items regardless of their location. What matters is the Document ID itself instead of the link.
The Document ID links update themselves when the daily Document ID Assignment job runs after you change the name of the subsite, so that they continue working in the site.
Currently, there is no tool or mechanism to maintain the Document ID links.
If you are looking for a way to keep the original links, you can refer to the article here for methods on URL redirect: List of useful SharePoint redirect rules for URL Rewrite.
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