We are also looking for this feature.
Specifically, the chat forwards a transcript to the chat-ee. If they reply to the transcript via email, it comes into the conversations inbox, but there was no way for me to easily forward her inquiry - as the person who handles her question is not a user in HubSpot.
A workaround I found was to reply to the email, but delete the chat-ee and input the email address of the person to whom it needs to go.