So far as I can tell, Korsgaard's view in self constitution (I'm about 3/4 through it) is something like: action is constitutive of agency insofar as our capacity for self-directed movement against the background of certain perceptual and environmental constraints depends: first, on the freedom we have as agents to formulate conceptions of the work and our place in it and see those conceptions as binding for us, and, secondly, to identify something (movement, internal utterance, mental or physical behavior or performance, etc) as an action is to say that it contributes to the maintenance of our being or form -- it's what it means for us to be the kinds of animals that we are.
That's convoluted (both the original view and my description of it), but I think Korsgaard is on to something.
In my view (which is shared by a number of theorists who have so far been unable to articulate the view in a fully coherent, let alone persuasive, way), the essential link between action and identity is narrative, which I'm here defining as something like: the descriptive whole into which all the actions of an individual fall in a systematic and intelligible way. The idea is something like: we are constituted -- as Korsgaard thinks -- by our actions, but these actions are only intelligible in light of some narrative that encompasses all such actions into one intelligible, coherent whole. To be a whole self, a complete self, is to possess the authoritative self-knowledge of that narrative, and to be able to act in accord with it (and endorse it fully) into the infinite future.
In this way, then, for Korsgaard and for me: (1) Selves are constituted by authoritative actions of the individuals whose selves they are, (2) these actions are not wholly free or unconstrained.
For Korsgaard, the actions are constrained by one's natural form (rational animal, etc), and by some sort of survival / reproductive imperative (among other things, like more imperatives). For me, the actions are merely constrained by the extent to which they constitute intelligible parts in the whole of the narrative of one's life / self.