Networking without feeling pushy after 5 years as a project coordinator

I sent 5 messages on a professional network and got two replies. Any experience at all would help. I want advice, not a job directly. What did you ask?

Follow up with a thank-you and a short update. One of my chats led to a referral a month later.

Thanks for writing this out. I went through something close to it last year, and it settled down once I kept a simple log of every call and form.