I'm experiencing similar behavior / related problem on Samsung Galaxy Edge running Android 6.0.1. None of the messages sent - either from phone nor desktop appear in the thread. I mostly struggle with the absence of previously received replies not appearing in AirDroid. As a result, I'm seldom using texting feature from the desktop.
To solve the issue for you more quickly and efficiently, would you please report this issue via Feedback with AirDroid on your device?
Steps: open AirDroid on your phone-->Me--> Settings-->About--> Feedback.http://forums.airdroid.com/viewtopic.php?f=4&t=28318
Please do tick the box of “Attach error logs” so that our developers can check the issue for you. It would help us to gather the troubleshoot with more details (there’s no personal data collected), we shall be able to prioritize your request.
Should you have any further questions, always feel free to let us know anytime!