There are a few services that move files between clouds. They're usually paid, but I think they have some free functionality: https://www.jolicloud.com/, https://mover.io/, https://www.multcloud.com, http://www.cloudsfer.com/ are the ones we know. Joli and Mover recognize put.io, 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. ).
But it's never a good experience. Constant timeouts, etc.
Will we ever do it ourselves? Maybe. We're not sure. Past behaviour 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.