Getting email from 1 hosting company to another hosting company

Helping my brothers who are very low tech.  

They have their own domain. Hosted at a shared hosting company. They check / send their mail ONLY from the webmail of the hosting company.

They are moving the website to another shared hosting company (and setting up email at that new company also I would think).  They are asking me about getting / saving all the the old emails on the first hosting company.

I know I could use my outlook, set up a POP account and download the inbox for each account.  But other folders?  Sent?  

Maybe I'm not much more techy than them - should I really set up outlook for IMAP?  then I can export ALL the folders into a PST?

Next issue - they don't use / have  outlook. Can that PST of inbox / sent / etc be pushed up to the new server? (set up imap to the new server, and drag /drop emails from the PST into the IMAP appropriate folders?

If that won't work, then what would you recommend for someone that is using just the webmail of the hosting company but want to see the old emails once in a while?  Thunderbird for the old emails?

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Setting up IMAP then importing it to PST is the right idea and that's what I would have done.  I haven't used Thunderbird in ages but you can supposedly import PST files into TB.

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Thanks!!Can I push the emails from the pst up to the server through imap?

Much prefer it all in 1 place

In a separate incident, I tried pulling emails from an iPhone by dragging into an imap account but that didn’t work
If you use Outlook you can import and export PST, to and from.  Not sure about iPhone as I normally use Android and BB.  In Outlook you can drag and drop to and from pretty much any folder.
nociSoftware EngineerCommented:
if there is IMAP on both site why not use imapsync:

should be able to run on Unix/Linux and CygWin on windows.
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