Downloading e-mail from a master mail server

[Marc Roos]

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

Why don't stay at exchange? I have been testing a redundant setup with 
2016 for a while and it is do'able. If you are not using public folders, 
you have lots of clients that interact quite well with it. Eg. macos 
mail syncs tasks and notes with the native applications remindes and 
notes.

 >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.

Get one you do rely upon. I can't imagine you do doing a better job in 
most cases if you do not know dovecot, do not read the manuals here, do 
not have a test environment. Because you do not have the 'resources'.

 >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.

Yes the pop connector in exchange 2003. We have one running still in a 
firewalled environment and limited l2tp vpn access. Maybe you can switch 
this to some spla(?) licensing program of ms, so you pay per account?

 >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.

Your provider should be able to facilitate you with a vpn gateway, to an 
'offline' exchange server, that is not really a big deal. 

 >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.

So configure your pop connector not to delete after download. You will 
still have the problem with users creating online imap folders that the 
popconnector cannot see. Maybe you can solve this with an imap sync 
program. But this is a wrong direction to take.  

 >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:
 >

Configure clients to store mail files on the a network share, and do 
backup on the server? But also this is thinking in a wrong direction. 


 >
 >So I am trying to design a solution with Dovecot, but I know too 
little 
 >about mail servers. 

So study it, read about it. How can you select a good service provider, 
without this knowlegde?

 >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.

If you have to ask this question, you should not be doing this. 
Especially if you value your security so much, as you mentioned before.


 >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.

Here is described how you configure dovecot.
https://wiki.dovecot.org/


 >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?
 >

If you do not have time to read, get some help. Otherwise you will 
create a mess for your users. The problem with IT is that everyone is 
just jumping in without education nor responsibility. 
It is like your dentist is earning something on the side in the weekend 
as a brain surgeon.
Start acting like a pro