I've tried backing up all of my messages but I can only find how to back up the sms messages in the default message app, but nothing seems to detect the messages that have switched to the "chat" type.
For the moment, chat is SMS. If you back up your SMS you should be backing up chats too. (But remember, Chat is a standard, not a Google app, so we may see different carriers implementing it differently at first. It's going to take a while before Chat [also called RCS] is fully standardized.