Hi,
Messages on the Mac will only sync iMessages from the time the iPhone and a Mac are linked by adding the Apple ID to the iPhone and then the pop up accepted on the Mac to add the iPhone number to the list of Alternatives.
This means that a New Mac will not get older iMessages from the server (They actually no longer exist).
iMessages are "duplicated" at the iMessages Severs and sent out (pushed) to the specific devices.
Once that device has received the iMessage the iMessages is deleted from the server
This may mean an iPhone has an iMessages and that "version" has been deleted from the iMessages server - whilst a version that is going to be sent to the Mac is waiting for the Mac to respond so it's version can be sent.
iMessages are Pushed as it is called from the servers and not "fetched" like email are in the Mail app.
So...
A New iPhone will not get "old" iMessages that still show on a Mac.
If the Mac is New (or the Mac User accounts is new) and the iPhone has the "old" iMessages the Mac will only see/sync the "new" iMessages.
There are ways of using iTunes to Save a Backup of the iPhone and tools that can be downloaded to "Save" older Messages to the Mac if they have not synced in Messages.
However there is no method to get older iMessages on the Mac to put them on an iPhone
8:59 pm Friday; May 1, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
