Duncanopsammia do not 'require' feeding. They are photosynthetic. However, they do grow faster if fed routinely. I never target feed the colony I have in my display tank.
As duncans are lps, they grow in a hard skeleton, so they do not 'move'. If your duncan is moving, it either experienced polyp bailout, which I have not heard of in duncans, or you have something else, most likely an anemone of some kind.
One thing that seems to help duncans get longer skeleton, so they can be fragged easier, is more flow. A duncan in less flow will sprout out babies on the sides of adult polyps, but remain in a tight bunch. Duncans in more flow seem to stretch more, and develop more of a branching look as the new skeleton grows. Thats what seems to be happening with my frags anyway. Either way, the skeleton grows slowly, IMO.
Hope that helps.