From what I am reading it is NOT the fact of the computer change (Mini to Pro) but rather the OS X change to Yosemite (you are not running mavericks based on the style of the mail window). So my best guess is that the mail database you are coping is formatted differently between the mail systems.
SOLUTION: All (that I know) Email systems store their email data on their servers so if you delete or remove (or just move to the desktop) the 'V2' folder your provider should send all the past data to your computer. If you don't want to use this solution you can also just check your mail in your browser.
EX: johnappleseed@gmail.com
Every email john has ever sent or received has been processed by gmail.com.
So if he has any problems he can visit mail.google.com and log in to directly use and manage his account.
I have struggled with apple's iMail (on my MacBook Pro late 2009 running OS X Mavericks, soon to be Yosemite) in the past and I personally thing they did a poor job and revert to just using the website or using their iOS version.