Hi @anon49395950,
As @Stephanie_Oberg quite accurately described, Asana only shifts dependent task if (a) you’re in Timeline view and (b) the dependent task conflicts with (overlaps with) its shifted dependency.
If you want to have all dependent tasks auto-shift, regardless of overlap or not, regardless of whether you’re moving forward or backward, and regardless of what view you’re in - that capability is available via the Auto-Adjust Workflow type of my Flowsana integration. It will do exactly what you’re asking for. 