You will need
  • - MS Outlook 2007/2010,
  • - access the old and new mailboxes.
If you decide to switch to Gmail or Outlook - you're in luck! Go to settings and add the old mailbox via a special menu. After some time, the mail will be downloaded itself.

If not, the transfer will have mail through any email application like Outlook or The Bat. Since the first program is more common later examples, we will use it.

So, start Outlook and click "Next" 2 times.
Mail migration
Enter the name and address of one of the mailboxes. All other settings of Outlook will likely be able to do myself.

But if Outlook was not able to determine the settings yourself, find their "Help" section of your email service (links at end of article).
Migrating email from Outlook
Now it remains to connect the second mailbox. Go to "File" - "account settings" - "Add" - "Next". Plug-in Inbox similarly as in step 2.
How to move mail
Like transfer a file from one folder to another, move the old Inbox to the new one. Don't be surprised if the Outlook will be very slow - this is normal, just wait some time (perhaps several hours). In the end, all emails will be migrated to the new mailbox.
Mail migration Outlook