It is really no mystery as to why a chain would move across the chainrings. It all depends upon the position of the front dérailleur cage in relation to the chain. Front derailleurs are designed with a spring that always moves the derailleur to the smallest chainring if tension is released on the cable. It only moves if more cable is released. That has to be what is happening in your case. It takes a fair amount of force to overcome the action of this spring. Try moving it with your hand.
All of my trikes have had triple chainrings which were actuated by a friction type bar end shifter. There is a way to increase the amount of friction on the shifter so that it stays in gear. If you look at the shifters shown here you will see the screw heads in the left side of the picture. You tighten those just enough so that it does not slip and so that you can still change gears easily. With bar ends, it is not rocket science. If that does not work, then you need to look elsewhere to see what could be releasing cable to make the derailleur move.
Here is a clear picture of the bar end shifters I use
https://www.bigandtallbike.com/Shima...rs_p_9389.html