There are a few services that move files between clouds. They're usually paid, but I think they have some free functionality: https://overlake.io, https://www.dissident.ai/, https://mover.io/, https://www.multcloud.com, http://www.cloudsfer.com/ are the ones we know. Overlake and Dissident recognizes put.io, and for the ones that don't, you can use FTP or WebDAV as source (Connect to ftp.put.io or webdav.put.io (via https://) with your credentials. Here's an article about that. ).
If you are comfortable with the command line, you could also use rclone. Here's an article about connecting rclone with put.io:
Will we ever do it ourselves? Maybe. We're not sure. Past behavior of cloud services makes it hard to risk many hours of development time. We don't want to wake up someday and see that they have blocked access to their API.