As of v6.2.0 Dux-Soup includes a Message Bridge. It's simply enabled via Dux-Soup options -> connect:
Once enabled, Dux-Soup will read and forward all messages from the Standard LinkedIn in-box to the active webhook. It does this by checking your LinkedIn Inbox for Unread Messages, every 15 minutes. Please note that this functionality requires you to leave your Chrome browser active.
For details on the supported LinkedIn message types and the Webhook Event structure, see this article.