Dux-Soup Turbo 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 and Sales Navigator inbox to the active webhook. It does this by checking the inboxes for new Messages, every 10 minutes. In addition, Dux-Soup will check your network for new connections and generate and "INVITATION_ACCEPT" Messages for each new connection found. 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.