It’s not possible in Asana to have a dependency shift the date of a dependent task in another project. Dependency-shifting is only available in the Timeline view, which only covers one project.
There are two possible solutions I can suggest:
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You can multi-home the dependent task so that it lives in both projects; that way, when you shift its dependency, the dependent will be in the same project and its date will shift.
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You can use our Flowsana integration, which does dependency-shifting regardless of what project the dependency and its dependent are in. (However, I do have to caution that building a workflow where dependency relationships span multiple projects can sometimes be challenging for users to work with. It just introduces extra complexity and less visibility to the overall workflow.)