Switching Email address?
Switching hosting-providers?
Gigabytes of valuable mail you cannot afford to lose?
You could use free THUNDERBIRD Email program and load it with more than one IMAP Email account, for each Email system you have
Then drag files and folders between the two- effective for low volumes but highly unreliable and requires a high level of attention to detail, double-checking and second guessing as it is you alone that keeps track of what you have and have not copied- The rate is typically about 30 standard sized messages per minute- Attachments can slow down or break the transfer : (

Best Strategy-
First:
Backup your Email somehow, do some Housecleaning- Sorting messages by attachment size and try to kill of the largest offenders– you can detach important attachments like photos THEN delete the messages- Thunderbird even allows you to “Detach” attachments leaving the parent email behind without its attachment,

Sort By Subject
This will bring all the Mailing-List items into a group as they tend to share the same initial wording at least like 
“[Poodle Fanciers] Q: About miniatures…”
Sort by Sender. 
Whoever sends you all this cartoons everyday can be batched up and dealt with. Finally:
Empty Trash.

Art Of Logic specializes in large IMAP mail migrations using PERL scripts customized from the command line using
https://imapsync.lamiral.info by Gilles Lamiral. That site actually will do up to 3GB free at one go for you
(YOu are of course trusting your Username & Password to him while doing so 🙂 I use highly customized version of this script whose only size limitation is the transfer rate and time

We can move a whole office of Emails between hosts for you or between providers

In particular we can migrate you off ugh Office365, Yahoo!, AOL.COM and other vexatious troublesome hosts. 
It seems like you may have Downtime on your valuable communications– Not necessarily.
We can sync up your Email folders from our high speed colocated servers without interfering with your Workflow.
In fact, we specialize in moving everyone’s Emails into the likes of GMAIL, Google Apps AKA Google Workspace
Usually $5/mo per account, this is free if you are a legitimate NonProf 501(c)(3) for your whole enterprise. 
We can do this for you, too.

Enjoy!