LinkedIn messaging is a very powerful way to fill your live events and webinars.
Focus on starting conversations with short messages. The best way to get someone's attention and start a conversation is to use Dean Jackson's 9 Word Email approach.
Be curious and authentic when you engage your LinkedIn connections. I start with a Yes or No question to get them interacting and gauge their interest in conversing.
Ask open ended questions so they give you details after the conversation is flowing.
Never try to sell them in the messages.
As you are closing the conversation, ask them if they would be interested in learning about your live event or webinar.
If they show interested send them a link to the event or webinar page.