I'm a novice user FYI. My messages used to be synced so that the conversation kept going on iMac iPhone and iPad mini. Then perhaps after a trip to Alaska it stopped doing that. I'm not 100% sure if that's when it stopped. I've noticed that my iBooks content syncs between phone and iPad but not iMessages. This is very inconvenient. All three, actually I have 2 iMacs so 4 devices are with same Apple ID. THANKS!
All syncing of messages is done with your iCloud account. First check the iCloud settings on the working devices and then compare them to the settings on the non-working device. If they are the same you could try removing and re-adding the account.
Note that Messages on iPad is only going to show you conversations with other IOS devices. So if you are texting with someone on a non-IOS device (say Android, Windows Phone, Blackberry, etc.) and the text bubble is green then it won't show up on the iPad (I believe).