Here is how I am doing it, if it's good or not, for any one else I don't know.
My mail (imap) is on my own domain, when on computer I use the webmail function with my web browser at my web hotel (one.com). I keep all mail that I want to keep on their server, the rest will be deleted, I never download the mail to my computer(s).
On my phone I use K-9 (no google shit). K-9 is set to normally only downloading the last 25 mails but I have full access to all my mails always from the phone if I want to.
Even if you download your mail to your pc you can choose to let them be on your web hotel instead instead of deleting them when downloading them, or?