Starting on the very first bar, the alignment for 1/4 is different than 2/5. Therefore, the slant of the hand should be different as well as the angle of the wrist. If you hold the hand and wrist still, you force only the fingers to move. This is not the most efficient way to play these thirds.
Also, since the hand is so closed throughout these passages, I've begun doing exercises to condition the inner hand muscles, the ones that allow you to make gang signs. I touch the tips of the 2/4 together, as well as the 3/5. This is so that it will feel more comfortable to keep my hand so closed.
As well, I do not press directly down with the fingers, but instead, use the pinching motion (e.g. pinching 1/4 together) to help depress the keys. This is much more efficient but is a very small movement.