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I figured it out:
/variations?include=field_images.field_media_image&fields[file--file]=uri,url
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Short answer: yes, you can get Drupal to associate captions to Media Library images.
Longer answer: there are at least two different angles to approach this from.
Approach #1: Media entities are fieldable just like any other entities, so you can add a Caption field to the Media entity itself.
Go to /admin/structure/media/manage/image/fields and click "...
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