How do I export emails I’ve downloaded into my live mail account ?

I'm moving into a new hosting and I'm stuck at the email backup. Here are some info for both hosting (only related info mentioned) :

Old hosting :

Domain : mydomain.com

Control Panel : cpanel

Mail : cpanel mail server

New hosting :

Domain : mydomain.com

Control Panel : none

Mail : Outlook live mail

I need to backup the old emails from old hosting into outlook. I have create live account (this account i'm using to post this) as administrator, added custom domain, verify domain ownership and insert members, which are going to replace the accounts from the old hosting. I will explain my problem in this scenario :

account *** Email address is removed for privacy *** is a new live mail account, member of its admin account (this account i'm using to post this). To backup, I need to create send-and-receive account that "pull" all emails from *** Email address is removed for privacy *** in old hosting. I have configured the mail address, protocol, username and password, and for a short time, it has successfully pull some of the email until this problem shows up. No solution, it just stopped pulling emails. I waited for 6 hours and I only got 30 emails. The total email was 65 with size around 130MB. Some of the attachment has big file, around 3-10MB

The problem is : I have several other user I need to backup and they have thousands of emails with size around 7GB total. Some of the emails has 70MB attachments. How long I have to wait until they all finished ? 

So, I tried other method, which is downloading the entire emails into Windows Live Mail (application). In WLM, I create an account, which is the *** Email address is removed for privacy *** from old hosting. It was faster of course, with no more that a minute, all entire 65 emails along with attachments are now at my PC storage. So my question is how do I "upload" these emails and attachment I just downloaded into an account in WLM into the new *** Email address is removed for privacy *** ? I want them to be uploaded into "backup" folder. I want these emails are there when I log in as *** Email address is removed for privacy *** in mail.live.com

In short, I want all emails from my previous hosting mail server copied into new account using live mail, both has same address (*** Email address is removed for privacy ***). Please tell me if you have better solution. 

Thanks in advance.