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You can drag to a menu; you can drag to the dock - but can you drag to a Dock Menu? eg to open a document different ways based on which item in a dock menu you drag to.

I don’t believe so, no.

Since when can you drag to a menu?

Look at this:

It uses Accessibility API to locate the position of its icon on the Dock, and then draw a floating window which looks like dock menu. By the way, how can it use accessibility API when I disabled “access for assistive devices” in System Preferences?

Use AXMakeProcessTrusted or

sudo chown :accessibility myApp sudo chmod g+s myApp

in Tiger.

If you must, then authenticate and use a small helper program to do the dirty stuff, but don’t make the gui run as superuser..