How to move mail messages between email accounts using an email client
If you want to copy your emails from one email account to another, you can easily do this with an email client (e.g. Thunderbird). You can move the messages from any folder. You can either move the messages from one account to the other deleting the messages in the old account, or you can create a copy of the messages so that you don't delete them in the account from which you want to copy them. You can do this with pretty much any email client.
The first thing you should do is to add the email accounts in the email client that you want to use. For more information on how to configure email clients to connect to email accounts you can refer to the email tutorial about the particular application or you can check the shorter article version. Use IMAP for incoming mail server. After you add the email accounts you can manage the inboxes of both.
One way to move messages is to drag and drop them. First click on the inbox folder to view a list of all the inbox messages. Then with your mouse highlight the message(s) that you want to move. To mark all the messages, highlight one and press Ctrl and A on your keyboard. Then press and hold the left mouse button, and drag and drop the messages in the inbox folder of the other email account (the folders for all accounts that you have added should be listed on the left side of the email client). In this way the messages will be moved (and deleted) from the first inbox folder to the inbox folder for the second account.
The different email clients have different settings, but when you highlight messages with the left mouse button and you press the right mouse button you'll be able to select a Copy To option from the menu that appears, and you can select the inbox folder for the other account. This will copy the messages from the first account to the second, and it will not delete the messages from the first account. This function is offered by most email clients, though some might have only a Move To option that you can select, and some might have both Move To and Copy To.