Move large mysql table data from one server to another
You could use a cron job to connect to the existing database, make a backup of the table, select the rows you need to move (over one week old) disconnect from the sql server,  connect to the new server,  insert the rows you have previously selected using a loop,  disconnect from the second server,  reconnect to first server,  and delete the rows you don't need anymore. 

That's one way of doing it,  I don't know if it's optimal,  but it would work,  do not forget to plan the cron job to run Only when the first server is at its lowest usage to make sure you don't impact the performance of the live server.