If you have two remote locations in Transmit 4, say an S3 and an FTP location, and you move a file between these two locations, Transmit will automatically download that file, and then upload it.
Where does Transmit 4 store these temporary files?
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You can find this out by yourself:
sudo opensnoop -n Transmitin Terminal
This will list all created/opened files, it shouldn't be difficult to identify the temporary files.
The opensnoop program is remarkably useful, but to answer the actual question in case somebody is in a quick bind, for Transmit 4 the location of the temporary files is:
/Volumes/<Macintosh HD>/Users/<current user>/Library/Caches/Cleanup At Startup/Transmit/server-to-server-B18AC7A9-81C6-4AAD-8F2C-6CD70E65D7FD/data-file-being-transferred
Transmit appears to clean up after itself within a second or two of the upload completing. The name and path of the folder implies to me that the system might clean it up if Transmit doesn't get the chance.
Update 26th July 2018: Latest version (5.1.5) appears to store a file at a time in:
/Volumes/<Macintosh HD>/Users/<current user>/Library/Caches/Transmit/[UUID]/<host server>/path/to/file
The files are cleaned up more or less immediately after uploading to the destination server (afaict) but the subdirectories seem to stick around for some period of time (such that I now have ~600 files in the Caches/Transmit directory. Perhaps this is used for some logging or history, etc.