I found the following method which allows for transfer of single files at a time:
From the local directory containing the fly.toml file:
fly ssh sftp shell
This will start an interactive session where you can navigate, push and pull files within the remote machine.
get FILENAME
will download a file into your current directory.
It would be nice to have a recursive option to download a folder with all its files more easily though…
To replace a file, if the PUT command does not overwrite, you can first fly ssh console, then rm FILENAME to remove the file. Then add the new file via fly ssh sftp shell again.