If you encounter problems using PHPMyAdmin to export large databases, you can perform the export from the command line instead.
First open up wp-config.php and take note of your database name, username, and password.
Then SSH into the application layer of your environment and use those values to run the run the following command:
mysqldump -u [uname] -p[pass] db_name > database_export.sql
For example:
mysqldump -u mightybox-1234 -pmysecurepassword wp_1234 > database_export.sql
You can also export specific tables from the database like this:
mysqldump -u mightybox-1234 -pmysecurepassword wp_1234 table_1 table_2 > database_export.sql