It sounds to me like the stand is losing connection with the iPhone, though without more details it's difficult to be certain. If you'll post the specific stand you're using, someone may be able to offer more suggestions. A couple of ideas to get started:
- Check is that you have the proper StandBy setting enabled. Go to Settings ➝ StandBy ➝ Display, and under "Turn Display Off," select Never if it isn't checked already.
- If that's not the issue, check if your iPhone has Low Power Mode on. If Low Power Mode is enabled (Settings -> Battery -> Low Power Mode). Your iPhone's screen will automatically turn off after a few moments of inactivity, so Low Power Mode needs to be off for StandBy mode to remain visible.
Otherwise, all I can suggest at this point is to try restarting your iPhone, if you haven't tried that already. Also, if the problem persists, see if you can tell if the iPhone is charging. That would provide information about whether it's completely losing connection with the stand.
Regards.