挪动 means to move or shift something, but with the nuance of doing it slowly, carefully, or with effort — often because the object is heavy, stuck, or needs precise repositioning. It is less generic than 移动 (to move) and more physical than 搬 (to carry); you might 挪动 furniture, a car in a tight space, or your body slightly. Unlike 移动, which can be abstract (搬家, 移动电话), 挪动 is almost always concrete and physical.