When you’ve successfully started your Live Activity, you receive an Activity with a pushToken. Send it to your push notification server, and use it to send remote push notifications that update or end the Live Activity.
Live Activity: How can I send 'pushToken' to my push notification server?
You can send it using HTTP, exactly how the HTTP content is structured is up to you.
Once you've sent it, your app needs to check to see if its changes (its not often, but it can), if so then you have to send the new one.