The answer is a bit complicated, I’m afraid.
Currently, it works essentially the opposite of that: if you move to the right, dependent tasks (whether top-level or subtasks) will shift but only if you move the predecessor such that it overlaps with a dependent. If you move to the left, there’s no shifting because there’s no overlap possible.
Later this month, Asana is introducing a new variant of dependency shifting where it will shift dependents regardless of left or right direction, or overlap/no-overlap.
However, I should also point out what I believe is a flaw in their new design: the new variant not only shifts dependents but also predecessors. I think that’s a mistake and not something most people will want; and in fact I’ll mention that the Auto-adjust workflow in our Flowsana integration only shifts dependents, not predecessors.