There’s some basic functionality in Asana to move things around in your My Tasks sections (from the My Tasks Guide):
Tasks with due dates that have been marked as Upcoming or Later will automatically promote through as they approach their due date.
When a task is marked as Upcoming, the task will automatically move to Today at midnight on the day of the task’s due date.
When a task is marked as Later, the task will automatically move to Upcoming at midnight seven days before the task’s due date.
Personally I don’t find due dates useful in prioritizing my work because things constantly get moved around and reprioritized. Early on in my Asana adoption I tried hard to use these dates as a way to commit to getting things done, but I couldn’t make it work and they became a waste of time as I’d constantly go through them and adjust them all.
I’ve found that taking the time to manually arrange all my tasks in My Tasks is much more powerful than relying on auto-promotion. My workload frequently has to change (support emergencies, escalations,and what I refer to as time-bomb tasks of things coming back from a vendor or user), forcing me to re-prioritize on the fly. So I set aside a little time in the morning to plan my Today section. I created subsections in the Today area of My Tasks that are kind of like a personal Kanban. I share the sections with my team so they can also see my work-in-progress and my planned order and anything I’m blocked on. I put all my work for my next 2 weeks (we’ve adopted Agile and work in 2-week timeframes) in Upcoming in the order I plan to work it. I only use the Due Date to mark out tasks that are truly time bound so I can promote their importance manually as the date approaches. I don’t use the Later section much anymore except for follow-up tasks where I put reminders for myself to check up on other people or do a quick thing I don’t really have to plan for. This is the one thing I rely on the auto-promotion in Asana for, they’ll bubble up to Today eventually the day I need to “follow-up”.
Here’s a pic of my My Tasks:
Not sure if this is helpful for you, but let me know if you have questions.