Aha! When the scene reloads, the old Dialogue Manager remains, in order to keep its Lua environment with all its variable changes, etc. But the newly-loaded speech bubble UI's continue button points to the new Dialogue Manager. Then the old Dialogue Manager destroys the new Dialogue Manager to ensure that there's only one. This leaves the continue button disconnected.
I'm publishing a Dialogue System patch today. It will include a fix for this. You won't need to do anything. The patch should automatically fix the issue.
The full version will be released on the Asset Store in a couple weeks. The patch will be on the Pixel Crushers customer download site. If you don't have access to the customer download site, please PM me your Unity Asset Store invoice number. In case you're using the evaluation version, I'll update that, too.