How to search iphone messages by date
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However I don't sync this way and don't know the names of the apps involved. The "Sync" is only "Display on all Devices" and does not involve anything else if you delete a message on one device it remains on the other The iMessages are "pushed" at the devices that have the Apple ID or iPhone and when the iMessages has been sent that is it. The "Sync" as Apple calls it was different before this time.
Just give PhoneRescue for iOS a try now. Step 2.
I really don't use iMessages on my iPhone enough to even try "Load earlier Messages". Part 2.
If you have synced Backup the iPhone onto the Mac using iTunes it is apparently possible to extract the messages from the Backup. Swipe down in the Home screen with your finger to bring up Spotlight Search. Enter the keywords or phone in the search box and the messagws will automatically update as you type.
It will not re-sync if you remove the iMessage.