The beginning-of-turn ability works just fine for me, but I have never seen the other abilities trigger when the support is destroyed, whether through the effects of a spell (like Demolish) or by matching it with other gems. For what it's worth, I have never had my Field of Ruin destroyed with an opponent's Land support on the board -- do the gem conversions only take place if Field of Ruin gets to destroy a Land first?
//Edited Title -Brigby