Stop Mail Downloading Messages

This morning, my OS Mail app suddenly started downloading thousands of old emails from my gmail and iCloud accounts, back to 2005. It has not done that before. How do I 1) stop the process; and 2) prevent it from happening in the future?

I've checked Mailbox Behaviors, Internet Accounts, and other settings and don't see how to do either.

I would like to keep some old messages in my Gmail account online, without downloading every one to my MacBook.

Thanks!