Many months after I first asked this question, I found the answer.
In a nutshell, the problem was that I was not piping a big asset when transferring it from one server to another. In other words, I was downloading an image from one server, before uploading it to a S3 bucket. Instead of streaming the download into the upload, I downloaded the file into memory, and then uploaded it.
I’m not sure why this did not show up as a memory spike, or elsewhere in my statistics.