You should not use phpmyadmin to export big tables from mysql. You may get partial exports because of php server timeouts.
Best way is to connect via SSH to the server and use mysqldump to export the dataset. Then you can download it and transfer it to the new server. There you could use the mysql command to import the dataset.
After the export you should always open the sql file with an editor and verify that the last line contains the following text pattern:
-- Dump completed on [date time]
A partial export will result in data loss if it does unnoticed.
Export from old database:
mysqldump --allow-keywords --opt -u USERNAME -p DATABASE > backup.sql
Import to new database:
mysql -u USERNAME -p DATABASE < backup.sql