MacBook – Stopping iMessage from displaying texts and iMessages on the MacBook screen

No, there will be no sync.
Only the devices, that are registered on iMessage at the point of time, when the message is sent by the sender, can receive the message (technically). The messages are end-to-end encrypted for each receiving device.
In other words: Your new device can't decrypt the old messages, as the original sender didn't know about this new device.

See this article:

You’ve actually got one set of keys for each device you add to iCloud, and each iMessage is encrypted independently for each device. So if you have two devices — say, an iPad and an iPhone — each message sent to you is actually encrypted (AES-128) and stored on Apple’s servers twice. Once for each device. When you pull down a message, it’s specifically encrypted for the device you’re on.

Apple's Security Paper

You need to turn off iMessage on all the devices that have it on. If you have iMessage enabled on iPhone, iPad, and Mac, turning one off will not disable iMessage entirely.

For example, if you enable iMessage on iPhone and Mac, but only turn off the one on iPhone, People can still send iMessage your phone number or email linked to your iMessage account on Mac.

Answering your question, the message will appear on the devices that has iMessage but not on the one you’ve disabled, however, if you turn off iMessage on all devices, then your friends will send you a normal text message.