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What about trying to get them to get the weight of the hand 'behind' the finger they are playing.So that when you go up and down say c,d,e,f,g,f,e,d,c it produces a rocking motion of the wrist.
You could ask them to practise watching their fingers to not collapse. Show them what the right way of the hand postion is like-noting specific features such as only your finger tip is touching the keys;not the entire fleshy part and the nice curvature that the fingers make.Then show them the wrong way that they are playing it.
Ask them to practise daily putting their hand with thumb on middle c, 2nd on d etc and practising going up and down noting their exact hand shape and the part of their finger that is touching the key.
hope this helps
good luck!
Sheena

Can't help with exercises, but as for the flat hand problem...
When my piano teacher was trying to explain hand position to children, she got them to cup their hand around their knee (...their own knee.. don't want to get you arrested here....) and then transfer hand, in same position, to the keyboard.
Perhaps you could ask them to do this at home each time they start to play, to remind themselves what their hand shape should be.
I found it helped to think of an eagle's claw (though they might just interpret that as a very stiff hand position, which I guess wouldn't help at all..)

A big thank you to Gil for the advice and help with the fingering for a smooth two octaves in the C major scale with both hands. I have finally managed to do it, albeit very slowly, following her advice.