With WinSCP you can manipulate remote files directly on the server without downloading them to local machine. You can particularly move or duplicate remote files to different remote directory. You can also duplicate a remote file to different server using local temporary copy.
You will be then prompted with the dialog, where you can alter a destination directory and specify operation mask.
You can also use the function to rename a group of selected files using operation mask. Just remove the target path, to make current directory the target.
Note that most servers do not allow moving files across different file systems.
To duplicate remote files, select them in file panel and then go to Files > Duplicate.
You can also drop remote files to any session tab to duplicate them.
Yet another option is to copy the files to the clipboard and paste them to another folder or even another session.
You will be then prompted with a Duplicate dialog.