And then there's something that begins with a white dusty-looking deposit on
the leaves, which turns even large, mature plants into nothing, in about 2
weeks. :-(

Ah ha the infamous powdery mildew, Normally attacks relatively mature plants.
Lots of controls, but baking soda works about as well as anything. I think
someone on the board has mentioned using milk, works but a lot more messy than
baking soda, Copper sulphate based products also work.