Why do the Powerlines switch off on their own?

This is normal, the powerline 1000 Mini automatically switch to sleep mode if no activity is detected. When a signal is detected on the Ethernet port, the Powerline 1000 Mini will turn back on. It may take a few seconds for a Powerline 1000 Mini in standby mode to detect a new activity and turn back on.
The automatic power saving mode reduces your consumption up to 85%.