Short answer: IMAP and the mbsync tool (aka “isync”). It can sync between two IMAP accounts or between IMAP and local storage (either/both ways).

If you just want syncing between two IMAP accounts there’s also imapsync, which is available both as a program and as an online service run by the guy who maintains the program, priced as “pay what you want”, which can migrate your inbox on the fly to another service.

What I’ve done myself is to run mbsync periodically (made myself a custom Alpine image with cron and mbsync) to bring emails over. Added an IMAP server container on top of the local copy of the emails (tons of options, Dovecot is popular). Added a webmail app behind reverse proxy, talking to the IMAP server on a private docker network (Roundcube). And a Borg Backup job to take an extra backup (incremental, deduplicated and encrypted) of the email archive.

In theory I could also connect the webmail to the SMTP of whatever email provider I’m currently using and be able to use it to also send. I don’t do that because I have email clients connected to the provider on both desktop and phone so it’s not a requirement, but I could if I wanted to.

This approach lets me periodically trim down the emails stored at the provider to only the most recent. This lets me also use providers that offer small amounts of storage. My recent emails are available instantly through IMAP to the provider. Starting within last 24h and going back forever they’re available at the archive webmail.

I can switch email providers at any time as fast as DNS records propagate (because I use @my.own.domains) and as fast as I can update the IMAP/SMTP credentials for my phone/desktop/mbsync.