No, cataracts do not move in the eye. Cataracts are in fact a clouding of the lens. The lens itself is a stationary part of the eye and only appears to shift as the eyeball itself moves around. They can develop on the eye at various parts but will not move once they have developed. Similar to the way that a scab on the arm will not move or shift until it is removed. This information was obtained after a small amount of research was done through Google.com, medical sites, and Wikipedia. Cataracts are easily treateable by modern medicine and are not a problem in most developed nations.