After moving overstuffed Apple Mail inboxes to "On My Mac", mail reappears in inboxes

I know I'm doing this the wrong way and could use some help. I have a LOT (100,000+) of mail in multiple email inbox accounts in my Apple Mail on my Mac to the point where it's causing issues with the app. Some are IMAP accounts, some are old POP accounts, and some are Exchange accounts.

In the past, I've occasionally created a new folder in the "On My Mac" area, copied everything into that folder, and deleted them from the inbox. The last time I did this, it seems they later reappeared in the inbox as if they were redownloaded from the server.

I believe moving emails to "On My Mac" is the recommended way to get emails out of the app, but I think I'm doing it wrong. I know with POP accounts, mail is download and immediately deleted from the server, but I think it's the IMAP accounts that are coming back. Could someone walk me through getting these off of the server and into my "On My Mac" folders? I know I need to get rid of these POP accounts. Thanks!