How can I save Gmail emails in Apple Mail?

I have a problem archiving my Gmail emails in Apple Mail.

In the past I have created mailboxes on the Gmail server where I can move messages I want to save out of my Gmail Inbox and so declutter the Inbox. Now I find if I move the message to the mailbox it remains in my Inbox. If I delete it from my Inbox, that also deletes it from the mailbox. Gmail's (or Mail's) behaviours seems to have changed, as I used to be able to archive messages in mailboxes without them staying in my Inbox.

How can I save a message and remove it from my Inbox? I realise I can use local mailboxes "On My Mac" where this does not happen, but the advantage of mailboxes on the Gmail server over local ones is that I can see them on any of my devices logged into my Gmail account.

Has there recently been a change in Gmail or Apple Mail that is causing this to happen?

brianbron

MacBook Pro, MacOS Catalina 10.15.7, Mail 13.4