Downloading e-mail from a master mail server

[R. Diez]

Hi all:

I am trying to migrate a small company from Microsoft Exchange / Outlook 
to Thunderbird. I am evaluating e-mail server software.

We have an Internet provider that we do not really want to rely upon. We 
also do not have the resources to maintain a mail server visible from 
the outside, especially regarding security updates.

Each employee has an e-mail account on our current Internet provider, 
and one e-mail account on our internal Exchange Server. Our Exchange 
server periodically downloads all e-mails from the Internet provider 
mailboxes to its local mailboxes. Therefore, employees cannot access 
their e-mails when travelling.

We have worked this way for years without problems. There are other 
ways, like VPN access etc, but we do not have the resources to install 
or maintain more IT stuff.

So I would like to keep this setup, with one important difference: 
e-mails on the Internet provider mailboxes should remain there for a 
couple of weeks. This is similar to Thunderbird's option "Leave a copy 
on server" together with "For at most xx days". This way, employees 
could access at least their most recent e-mails when travelling, if only 
over the Internet provider's web interface.

I could achieve such a setup without a local mail server at all, only 
with the Thunderbird clients, but I have not figured out yet how to 
automatically backup all mailboxes. See this question of mine:

https://support.mozilla.org/en-US/questions/1243605

So I am trying to design a solution with Dovecot, but I know too little 
about mail servers. How could I configure Dovecot / MTAs / whatever in 
order to achieve the "Leave a copy on server" together with "For at most 
xx days" mentioned above? This way, Dovecot does not need to be exposed 
on the Internet.

Failing that, could someone tell me at least how to configure Dovecot / 
whatever to download the external IMAP mailboxes to the internal IMAP 
mailboxes? When I read about Postfix and the like, I see SMTP and e-mail 
queues, but that's not what I need.

This guy wants a similar setup:

https://serverfault.com/questions/500591/fetch-all-mail-from-an-imap-server-with-postfix-to-distribute-on-local-imap-serv

But I did not understand the Fetchmail part, which seems to be 
discouraged anyway:

"Fetchmail can be used as a POP/IMAP-to-SMTP gateway for an entire DNS 
domain, collecting mail from a single drop box on an ISP and 
SMTP-forwarding it based on header addresses. (We don't really recommend 
this, though, as it may lose important envelope-header information. ETRN 
or a UUCP connection is better.)"

I also haven't understood yet the backup part with Dovecot. There is no 
central e-mail database like in Microsoft Exchange, right? How do I 
backup all mailboxes for all users? I probably need to stop the Dovecot 
server an any MTAs before backing the raw files underneath, right?

Many thanks in advance,
   rdiez