You don't seem to be getting specific help and I cannot give you anything definitive either, but will offer a few thoughts.
I have been using Luminair version 2 on iPad 2 and iPad3 with a DMX King basic artnet interface and a few different generic routers. It has worked very well with no issues for a year or so.
Your set up is different mainly because of the EZ Kling interface, so I would focus your investigation on that device. It is possible there is some software conflict between the standard ArtNet signals Luminair sends and what the EZ Kling does with them.
The specs for the ArtNet converters I have researched generally say that they will continue to output the most recent DMX information transmitted. So with my DMX King interface, if the Ipad and/ or router loose connection, the scene stays the same. In your case either Luminair would have to be actively putting out a signal to turn the light off, which I think is unlikely, or the EZ Kling device is doing it.
I doubt is is a router problem.
You might check what all is running on your iPad. Put it in airplane mode and turn off everything (like bluetooth) except WIFI. There is also a Luminair setting that says "always run in background" that you might try.
Good luck and let us know the final solution if you can.
