Hello CharleyM8859,
Are you using Messages in iCloud? Using Messages in iCloud would sync all of your conversations across your Apple devices. Learn how here: Use Messages in iCloud
Use Messages in iCloud
iCloud can keep your entire message history updated and available on all your devices, even when you set up a brand-new device.
Here's how it works
When you turn on Messages in iCloud, your message history is updated automatically on all of your devices. That way, you always have the same view everywhere you use iMessage. When you delete a message, photo, or conversation on one device, it’s removed from all of your devices. And since all of your attachments are stored in iCloud, you can save space on your device.
You can use Messages in iCloud on your iPhone, iPad, iPod touch, Apple Watch, and Mac. For your privacy, Messages in iCloud is end-to-end encrypted, which means you can't view or access Messages online via browser.
Learn more about end-to-end encryption and iCloud security.

Turn on Messages in iCloud
To turn on Messages in iCloud, make sure that you've set up iCloud and are signed in with the same Apple ID on all your devices, and that you've turned on iCloud Keychain. You also need to use two-factor authentication with your Apple ID.
On your iPhone, iPad, or iPod touch
1. Go to the Settings app.
2. Tap your name.
3. Tap iCloud.
4. Turn on Messages.

On your Mac
1. Open Messages.
2. In the menu bar, choose Messages > Preferences.
3. Click iMessage.
4. Select the checkbox next to Enable Messages in iCloud.


Turn off Messages in iCloud
When you turn off Messages in iCloud on your iPhone, iPad, iPod touch, or Mac, you can choose to turn off Messages in iCloud for just that device. Any device where Messages in iCloud is still turned on continues to store what you send and receive from that device in iCloud.
You can also turn off Messages in iCloud on all of your devices.
On your iPhone, iPad, or iPod touch
1. Go to the Settings app.
2. Tap your name > iCloud.
3. Turn off Messages.
On your Mac
1. Open Messages.
2. In the menu bar, choose Messages > Preferences.
3. Click iMessage.
4. Deselect the checkbox next to Enable Messages in iCloud.
If you turn off Messages in iCloud on an iPhone, iPad, or iPod touch, your message history will be included in iCloud Backup. If you set up an Apple Watch for a family member, Messages in iCloud is the only way to back up messages.
If you're already using Messages in iCloud, then it should be syncing across your Apple devices. This service would be an all or nothing sort of thing. You would need to sync all messages in order to use the service.
Let us know if that helps to sort this out, take care.