It's not clear exactly what you're trying to accomplish, but I'll take a couple of guesses and see how close I get.
If you're wanting to let rsync magically handle the difference between your local dump file and a remote dump file, you could just replace the
&& so that if
mysqldump finishes without an error, then rsync will transfer the differences after the backup is complete, and if
mysqldump fails, the synch won't happen.
On the other hand, if you're just wanting to transfer the dump file across the network and drop it somewhere other than the local machine, then this is an option:
mysqldump -u root -p database_to_backup | ssh firstname.lastname@example.org 'cat > /var/www/db_backup_file.sql'
...where /var/www/db_backup_file.sql is a path on the remote machine.
Or if you wanted to save it both locally and remotely at the same time:
mysqldump -u root -p database_to_backup | tee db_backup_file.sql | ssh email@example.com 'cat > /var/www/db_backup_file.sql'