Don't use phpMyAdmin to import large files. You'll be way better off using the mysql
CLI to import a dump of your DB. Importing is very easy, transfer the SQL file to the server and afterwards execute the following on the server (you can launch this command from a PHP script using shell_exec or system if needed) mysql --user=user --password=password database < database_dump.sql
. Of course the database has to exist, and the user you provide should have the necessary privilege(s) to update the database.
As for syncing changes : that can be very difficult, and depends on a lot of factors. Are you the only party providing new information or are others adding new records as well? Are you going modify the table structure over time as well?
If you're the only one adding data, and the table structure doesn't vary then you could use a boolean flag or a timestamp to determine the records that need to be transferred. Based on that field you could create partial dumps with phpMyAdmin (by writing a SQL command and clicking Export at the bottom, making sure you only export the data) and import these as described above.
BTW You could also look into setting up a master-slave scenario with MySQL, where your data is transferred automatically to the other server (just another option, which might be better depending on your specific needs). For more information, refer to the Replication chapter in the MySQL manual.