How do Mail Rules Work?

If your Mac is powered off, nothing happens, mail remains on the mail server, no rules run, as nothing runs.

If you want the rules to run while your Mac is sleeping or powered off, you will have to use rules that run on another client that is booted and running, or rules that run the mail server.

The Apple iCloud mail server offers this, as do some other providers.

If you’re running several mail clients each with rules running locally, the first mail client that polls the mail server and retrieves the arriving mail message “wins”.

You’re looking for the particular rules for the particular mail server. Some mail servers offer support for user-written rules. Some don’t. Details vary.

Here is some documentation for the iCloud mail server rules:

Set up rules to filter email in Mail on iCloud.com - Apple Support

How a mail server operates is a bigger discussion. And a discussion very much dependent on which mail server software.

Intro: https://www.samlogic.net/articles/mail-server.htm