Certain happenings in the world are transmitters of responsibility. For instance, if at t1 Helga sets a bomb to go off at t2, and at t2 the bomb goes off causing harms, the explosion of the bomb is a transmitter of responsibility: it transmits Helga's being responsible for the bomb being set into Helga's being responsible for the harms. At t1 Jim buys a pig, and at t2 the pig rips up his neighbor's cactus garden. The ripping up of the cactus garden transmits Jim's general responsibility for the pig's actions into a specific responsibility for damage to the neighbor's cacti. Transmitters of responsibility don't create responsibility ex nihilo: they simply take something that one is already responsible for, and work out one of the possibilities in that line of responsibility.
Some happenings in the world originate lines of responsibility, rather than merely transmitting responsibility from one thing to another. Barring further information, we would say that it was at t1 that Helga originated a line of responsibility for harms coming from the bomb (with three different kind of responsibility respectively for intended, foreseen and unforeseen harms), and it was at the very same time that Jim assumed responsibility for the pig's actions. I said: "barring further information." For we might find out that at t0, Helga had paid someone to hypnotize her[note 1] to set up a bomb at t1 (maybe she was afraid that when it came to the task, she would be too scared). In that case, the origination of the line of responsibility that included her setting off the bomb at t1 and the harms at t2 was in fact at t0.
One might be tempted to think that in the case of self-originated hypnosis, it as at t1, when she does the action of setting up the bomb, that Helga assumes responsibility for the harms. But this is mistaken as we can see when we compare the case to one where Helga hires someone to hypnotize Sally to set up the bomb. For it is when Helga hires someone to hypnotize Sally that she assumes responsibility for the harms of Sally's bomb. And why should this change if we replace Sally with Helga herself? (We can add an intermediate case where Helga hires someone to hypnotize the person who matches some description, but is unaware that she herself is the person who matches it.)
The distinction between responsibility origination and transmission is an important one. For instance it is the mental states of the agent at the time of the origination of a line of responsibility that are typically most relevant to ascertaining exactly how responsible the agent is.
Now, in principle, the compatibilist and incompatibilist could give exactly the same story about responsibility transmission. It's not an easy task to give a good story of how responsibility transmission works, and it would be good for compatibilists and incompatibilists to pool their mental energies to work on this task. The unavoidable difference between the compatibilist and incompatibilist will be in responsibility origination.
The starkest case of responsibility origination will be when the agent transitions from a state of not being responsible for anything to a state of being responsible for something, presumably at some point in childhood. But presumably there will be less stark cases later on. There will also be cases where we have at the same time both transmission and origination of responsibility. For instance, we might have an existing responsibility for a character trait, and the injection of new responsibility for that trait by a new voluntary self-identification with it.
The compatibilist is committed to the claim that either (a) responsibility origination can happen without any action or decision on one's part, or that (b) an action or decision can originate responsibility despite being the direct deterministic outcome of states one is not responsible for. If (a) is the story about the first case of responsibility origination, then presumably this is a case of internal and external factors that one is not responsible for causing one to be responsible for, say, some character trait or pro-attitude. If (b) is the story about the first case of responsibility origination, then the agent becomes responsible for a choice flowing deterministically from circumstances and mental states that she was not responsible for.
The incompatibilist will typically say that the first case of responsibility origination is one where one is not responsible for the mental state from which one makes the choice, but because the mental state leaves enough genuine options, and includes enough in the way of reasons for the different options, that allows the agent to originate the responsibility. Kane has a more complex story on which the first free choice is made from a mental state that one is compatibilist-responsible for, and that bootstraps one into a higher kind of responsibility. I think this is needlessly complex. There is no need to be responsible for the state from which one acts if that state allows enough flexibility and rationality.