Let me give an example.
Magus Primus has two filii, Secundus (the elder) and Tertius (the younger).
Tertius goes rogue, causing 100p in damages to a mage Quartus.
If Primus is alive, it's his duty to go after Tertius.
If he does so, and slays Tertius (recovering 20p worth of magical property in the process), then he has done his duty, no more and no less. The Quaesitores take the 20p and pay out 18p to Quartus, keeping 2p for themselves. That's not for calling the March, just an administrative fee, and Quartus knows better than to complain 
If Primus does go after Tertius, but one magus Quintus beats him to the punch, then he's still done his duty. The Quaesitores take the 20p, certainly keep 2p for themselves - not for calling the March, but for debating whether or not to shell out 2p to Quintus for his incovenience before paying the rest to Quartus. On the one hand, it's good if folks like Quintus are encouraged to go after outcasts, on the other, Primus was about to accomplish the same task for free, as per his duty. After a lot of paperwork, they end up paying Quintus 1p for the slaying, and 17p to Quartus.
If Primus is alive and well, and does not go after his filius Tertius, then he's violating the Oath. In principle, he can become an outcast too, and all his estate forfeit. At the very minimum, he can be asked to pay the minimum of all he owns and might come to own in the future, and the costs to slay the Tertius, to retrieve the 20p of assets he held, and to compensate the 80p left of damages (100p-20p).
Did I forget the quaesitorial fee? There's always a quaesitorial fee
Not for calling the March, of course.
All the above is obvious from the Oath. Ok, maybe not the quesitorial fee, but must we really dwell on that? But what if Primus is dead or in Twilight? That's the question raised in the passage we are debating.
It would seem that, since Primus is not around, nobody has the duty to go after Tertius. If someone does, slays him, and recovers the 20p in assets, those are due to Quartus, even though the slayer can fairly ask for a share in compensation of his role in retrieving them (not to mention in slaying a dangerous outcast).
But the Tribunal cleverly thought of a way to expand the duty to slay Tertius. Primus might no longer be around, but his inheritance is. And the duty to slay is attached to that inheritance like a curse. If Secundus has claimed the inheritance of Primus, it's his duty to go after Tertius. And not only that. If Primus' inheritance amounted to 50p, Secundus must: a) slay Tertius and b) compensate the aggrieved third parties with the 20p in assets Tertius held and all the 50p he claimed as inheritance. So those parties will get 70p (shared with the Quaesitors!) and Secundus will get nothing (but at least he won't be held accountable, as he discharged his duty). Secundus probably now rues the day he accepted the inheritance of Primus, but what did he know at the time?
If instead the inheritance of Primus amounted to 200p, then Secundus will be able to keep 200p-(100p-20p)= 120p once he slays Tertius. But if he does not, and some other slayer finally takes care of Tertius, said slayer can claim those 120p. Well, the Quesitors get their share, of course 