Moving files from folder to folder on a remote server using SSH without downloading those to local computer

I have a remote Debian test server to which I connect via SSH (Putty client). I wanted to move a lot of files on a remote machine to another folder:

remote: /mnt/a/ -> remote: /mnt/b/c/
RESULT@remote: /mnt/b/c/a/

I've used a move (mv) command connecting to a remote computer with SSH from my local computer:

mv /mnt/a/ /mnt/b/c/

I did that with about 700 MB of data (about 5 files) and it took forever to copy those files.

Does the mv command transfer the files to the local computer through SSH and then sends it back to another folder on the server? If so, is there any other command I can use to move the files only locally on the remote server?