The mysqldump involves the MySQL application reading the same data (not necessarily contiguously), and converting it to a text file of SQL insert statements (which is subsequently zipped, right?) so possibly that process involves temporary files, swapping, etc.
What sort of drive is it and what are the issues it's having?
mysqldump --login-path=local database| mysql --login-path=remoteThe new server will have to allow connections from remote IP which it doesn't by default (it listens on 127.0.0.1, but you can make it listen on 0.0.0.0 for all connections)