If the chart is in edit mode, you may move tasks along with the time line. If you keep “CTRL” key down while moving a task, it is moved together with all children/successors, otherwise it is moved within current possible limits. While moving, you may see the current EST (earliest start date) inline.
Child tasks can be moved inside the parent time frame. Maybe you need to extend the parent task length before you move its children. Successor is always placed after his predecessor task, so you cannot place him before. If you want to keep relative position of tasks while moving, keep the CTRL key down.