I still prefer to handle all my mails at home. I got backup set-up on my own and have anything in direct access. This network is connected to the internet via an end-user DSL connection without a static IP-address and without any useful SLA. However relaying mails over such an unreliable connection is still an unsolved problem.
There exists a lot of solutions for this problem. I'm currently building a new e-mail setup, so I tried to collect all the used and maybe possible solutions for the relay problem. I will show some of them.
The easiest solution is to just don't try to collect mails at home. This solution is somewhat against the rules, but I know people who prefers it. Access to the mails is usually done with IMAP and a copy is made if needed. This is not really a solution, but it works for many people.
SMTP is the mail protocol used in the internet. It is used for all public mail exchanges. By itself it can't be used to submit mails to remote destinations
With some preparations it can be used to relay mail over unreliable connections. There are three different turn commands in SMTP that can be used to start the mail flow. Two of them will be
Using SMTP themself is an easy solution. It can relay mails without much hasle But it needs either a fixed or authenticated connection.
SMTP can use a fixed connection. This is usually provided by some sort of VPN. The VPN can be encrypted, but does not need to. This allow the MTA to connect to the downstream server if it is available.
The other solution is to authenticate the downstream server. Authentication is available via TLS and X.509 certificates. It still needs some way to find it, but with dynamic DNS this is no real problem. Both variants can be combined with the extended turn command.
Extended turn allows to request a queue flush for a domain. It can be used to request mails only to be delivered if the downstream server is available at all. This reduces the load on the MTA.
On-demand mail relay is a rarely supported ESMTP feature. The Authenticated turn command effectively reverses the SMTP connection and allows, after authentication, the flow of mails from the server to the client via standard SMTP. There exists some stand-alone implementations, but no widely used MTA includes an implementation.
All e-mails can be delivered to mailboxes and retrieved by the end-users system and re-injected into the delivery system. Both fetchmail and getmail are able to retrieve mail from POP3 and IMAP servers. They are either directly delivered via a MDA like procmail or maildrop. Otherwise they are submitted via the sendmail interface or SMTP and delivered by the MTA.
Neither POP3 nor IMAP have support for meta data like the real recipient or sender.
The mails for each address are delivered to its own mailbox. This allows proper identification of the recipient address. It still have no real idea of the sender address. Because it needs to poll one mailbox per address, this highers the resources needed on both sides dramatically.
The mails for all addresses are delivered into one mailbox. The original recipient must be saved in a custom header to allow this information to be restored. Only one mailbox needs to be polled for mail addresses.
One of the oldest transports used for mail is UUCP. It effectively copies a file to a different system and pipes it into a program. UUCP can be used to transport mails in various ways.
Each mail is copied verbatim to the client. It saves the sender address in form of a "From" pseudo header in the mail itself and supplies the recipient on the command-line. So it have access to both sender and recipient address.
Batched SMTP transfers old-style SMTP transactions over UUCP The MTA (Exim supports this) or a special tool (in the bsmtpd package) writes this files and after a given time or size they are dispatched via UUCP to the remote system.
The bsmtpd packages was removed from Debian some years ago.
Dovecot support bi-directional synchronization of mailbox contents. It holds all data on both sides. The internal log is used to merge changes done on both sides, so it should not loose any data. This synchronization can be used to work with the data on both sides (via Dovecot of cause) or create backups. It needs shell access to the user owning the data on both sides.
There is no one size fits all solution for this problem. If you admin the remote system, you can implement any of this solutions If it is managed by someone else you need good luck.
Almost all solutions does not support current SMTP features. The one I'm really missing is DSN, aka configurable delivery notices. POP3 and IMAP handle already delivered mail and have no need for this. All UUCP variants does not handle it, because they are much older anyway. Only SMTP itself supports all of its features.
I still use rmail over UUCP for my mails at home and it works flawless. UUCP itself runs over SSH, it can compress data on the fly and authenticate using private keys.