Thanks for your answer @tbnv…
I do have access to the file system… How does it help me?

You need to :

1. Create the email accounts on the server
2. Download your email files and folders from the shared hosting
3. Upload that on the server

The files are often located in one of these locations:

/var/mail/
/var/spool/mail/
/var/vmail/

But it depends on the server configuration and can be located anywhere, actually. It’s important to note that the server configuration may be completely different. Not only about the main directory. For example the structure can be:

/var/mail/user_1@domain.com
/var/mail/user_2@domain.com

But it could be like this:

domain.com/user_1
domain.com/user_2

I can’t tell you more as I don’t know how the shared hosting and server are configured.

There is also a tool called imapsync but I never tested it. It’s quite easy to find tutorial about how to use it, like this one.

Maybe neither won’t work. I am surprised that your shared hosting allows you to access the complete file system, maybe there are some limitations.

The last resort may be the easiest one: transfer over IMAP. It’s a bit tedious but it always works.