Figures and their captions play a key role in scientific publications. However, despite their importance, many captions in published papers are poorly crafted, largely due to a lack of attention by paper authors. While prior AI research has explored caption generation, it has mainly focused on reader-centered use cases, where users evaluate generated captions rather than actively integrating them into their writing. This paper addresses this gap by investigating how paper authors incorporate AI-generated captions into their writing process through a user study involving 18 participants. Each participant rewrote captions for two figures from their own recently published work, using captions generated by state-of-the-art AI models as a resource. By analyzing video recordings of the writing process through interaction analysis, we observed that participants often began by copying and refining AI-generated captions. Paper writers favored longer, detail-rich captions that integrated textual and visual elements but found current AI models less effective for complex figures. These findings highlight the nuanced and diverse nature of figure caption composition, revealing design opportunities for AI systems to better support the challenges of academic writing.