This wouldn't happen to you if you had my patented dishwasher plate-o-matic!
And one can be yours for only $ 19.95!
Not really. And it's not really patented. Feel free to make one. It's just two scraps of plexiglass, two machine screws, two nuts, and four washers.
At least in my dishwasher, the flopping over happens when the plates in the front roll backwards, which then causes the plates in back to move out of the way by falling over, then get stuck in the sprayer arm for the upper rack.
I think this might be because my plates are slightly too large to be stable in the front row and naturally want to roll back, so probably not everyone experiences this as often as I do.
I don't use the heat drying cycle in my dishwasher to save energy, but I'm pretty sure the plexi should be safe. It's actually lexan polycarbonate plastic and it has a melting temperature in excess of 300°F, so it should be fine even in a heated drying cycle.