A user asked for help moving mail between two different email clients. He tried to use Outlook's builtin Import/Export commands but nothing was imported. The Import/Export commands were removed from Outlook 2013 and newer, eliminating this as an option in newer versions.
The easiest and free, but possibly the most time-consuming, way to move mail between any two email clients is using an IMAP account. Set the account up on the old client, move the mail into the IMAP account then set the account up in the other email client and wait for it to download. This can be used to move mail between Windows and Mac, or to or from Outlook. However, it can be slow. If you're in a hurry, use a utility such as Address Magic Plus.
If your current email host doesn't support IMAP, you can use a Gmail, Outlook.com, or Yahoo account to transfer mail between computers. It will be a slow process, especially if you have a lot of messages to upload (and then download).
To migrate mail using an IMAP account, add the IMAP account to your mail client then move the mail into the IMAP folders. Add the account to Outlook and wait for the mail to sync down. If you use a lot of subfolders, you may be able to copy the folders into the IMAP account. Do one folder at a time and verify the folders and contents synced up to the server.
Some IMAP servers have limitations on the characters supported in folder names and may not support deeply nested subfolders. Create a couple of folders in the account online so you know where the account puts the folders (under the Inbox or at the same level as the Inbox).
In Outlook 2013 and newer, verify the IMAP account is syncing all messages in File, Account Settings, double click on the account to view the Sync Slider setting. You'll want it set to All when you are migrating accounts or making a backup.
See Converting Outlook Messages and Calendars for other methods to convert email.