How to ensure you don't get auto routed back into closed conversations
If you've ever closed a conversation and then were immediately routed back in when the site visitor responded, it is possible that you are not closing and leaving your conversations.
Closing and leaving conversations will not fully prevent you from being routed back into the same conversation (the conversation routing will still follow your routing rules) but it will prevent you from being automatically routed in by default every time the site visitor starts a new conversation.
If you are only "closing" conversations, but not leaving, you will automatically join the conversation when/if the site visitor chats back in.
To close and leave a conversation in the old conversation view:
In your conversation view, click the "open" tab on the top right of the conversation, select "close and leave"
To close and leave a conversation in the new conversation view:
In the new view, you can use a keyboard shortcut to close and leave (⌘+Option+C). There is also an option to close and leave in the / command shortcuts.