Moving out of Cox internet area - how can I transport THOUSANDS of email to a different Internet provider

Is there a way I can download all of the emails to a thumb drive or hard drive back up system? I'm sure it won't mesh with the new internet provider (brighthouse road runner in FL) or do most internet providers "mesh" and work together. I need to move to FL to care for my Father and have set up an email account already on his internet acct. in FL- but the thought of moving THOUSANDS of emails one by one to the new ISP is daunting. I have been going thru them and deleting those that are not critical; but some are life altering emails - and deal with critical issues that I cannot "leave behind" no matter what - Does anyone have any suggestions? I thought about printing them all out and then scanning them - but was hoping that someone had another idea? To transfer/forward well over 4000 emails to my new ISP would be incredible and then open each once I get to Dads' and resort each one of them and place into new folders - that would be a full time job. Will people relocating out of necessity - there has got to be a swift and efficient way to handle such a process with all the technology out today Would love to hear any great ideas or processes anyone might have - thank you so very much!