In your My tasks, click on the Customize button in the top right, then Rules and create a Custom rule.
For the When, find the ‘When due date is approaching’ in the list of triggers. It will be greyed out so to make it available, you will need to adjust this rule’s settings (gear icon in the top right corner) so that it does NOT run on subtasks.
Set the trigger to due today or X days before the due date approaches.
Then your Do this card should be ‘move to section’.
Experiment with these to get your desired results. Note you will need to wait until midnight your local time for the rules to trigger.
Btw, if you use your My tasks in a List view, you could also try the Group button in the top right area and group by Due date instead of Sections, while clicking on the … and hide empty groups. That will create dynamic groups instead of Sections without the need of any rules.