iMessage to Mac when logged off Quick easy question - if iPhone iMessages are synced to Mac, then somehow you get logged out of messages on the Mac, would any iMessages sent by your phone while you were logged out on your Mac show up on your Mac when you log back into messages on your Mac? Or are those messages sent while you weren't logged in just on iPhone and there will be a gap in your Mac messages for the period you were logged out?
In my experience, iMessage syncing is not reliable but it does work most of the time. As a result, it's difficult to ascertain a definitive answer. The intended behavior is as you described it and there should be no gap in Mac message history. 

However, for me, it doesn't sync 100% of messages all of the time. It does sync most of the time. The other thing that can happen is a time delay. For example, I booted up my Mac after an extended period of having it off and no messages came in. The next day when I booted it up, everything came in.

If most of your messages are not syncing, I suggest disabling iMessage on your iMac and iPhone and then re-enabling it both places.