Changing the start date of a successor (linked task) via the edit Task Details pop-up in the task gantt or task list view, or via the start date field of the task list view will not work. This is because the task is held in place by the task link connecting it from the predecessor i.e. trying to change the start date in this way will see the start date snap back in place based on it's link lag duration.
NOTE: You can still use the edit Task Details pop-up to move the Linked Task via the task gantt or task list view - but this is via the Constraints tab - setting a Must Start On/Start No Earlier Than, or other relevant Constraint Type, with the Constraint Date.