I have iMessage on my Mac and iPhone.
I treat iMessage on my phone as the primary tool. Mac iMessage is convenient to send messages occasionally, but my Mac is not always connected to the internet.
A message has been delivered to my Mac, and after several hours it has not appeared on my phone. I have tried logging out of iMessage. Turning iMessage off. restarting by holding down home and power. waiting ten minutes and reconnecting.
Can't make it budge. I have had an ongoing conversation since then.