Spam on iMessage - What are my options?
According to Apple: "If you get an iMessage from someone who's not saved in your Contacts, you'll see a Report Junk link under the message." You then have to block the sender if you want to stop getting messages from them. 



But since you are asking the question I'm guessing that isn't showing up. And since the perpetrator is jumping to new Apple IDs every time you block them I don't know how useful it would be. 

If you do end up seeing the "Report Junk" link, maybe use it and don't block them, just mute the thread, reporting it every chance you get?