@ethan k If you send a chat or post through outbound to several users, it will not send the conversation via email. For those cases, you could set up the message as a Series
@ethan k my colleagues and I have recently been testing ourselves and it doesn't appear that Intercom offers this.
Breaking down what we found in testing - For chats, if your lead/user doesn't see your teammate's latest message (or series of messages) in the chat and that lead has left an email or the user is signed in to your product, after ~2 minutes, Intercom will send an email. From there users have a choice to reply in messenger or they can reply directly to the email. If they reply in messenger, the conversation continues there. If they reply directly to the email it turns into an email thread.
But there is no button to push to gives a user a full transcript of the chat conversation via email when it is closed. This is a feature that my company would love to see. Our users often request this.