On the Arbites front, they're more of an FBI sort of operation - they look at what threatens the operation of the Imperium, not "petty" law breaking - e.g. Murder.
Insurrection that threatens the Tithe: Arbites.
Inserrection that threatens the ruling House, but looks otherwise competent: Enforcers.
Insurrection that threatens the soul of humanity: Inquisition.
In terms of 40k, my inclination is more puzzle/mystery solving than strict notions of justice. But it's an old trope in crime fiction: solving the case isn't the same as making sure the perps are punished.
Uncovering the truth and unravelling the mystery is, perhaps, more what the Arbites judgements may come down to.
Highly dramatic, tense, bloody... But ultimately ends in "case dismissed" because its irrelevant at the Imperial scale. Everyone involved gets referred back to their own organisations with the suggestion of severe punishment for wasting His Divine Imperial Majesty's Servant's time.
I'm reminded, vaguely, why I lived the Calpurnia books so much - "Legacy" in particular.
In terms of lawyerly, legal, trial stuff, it's not a traditional guilt/innocence sort of substantiation, as much as settling in an agreed official interpretation of events. (And appropriate, multi-institutional response.)
Bloody heck though. I'm excited for it.
I'm also intrigued for the AoS part of the matter. I just recently listened to Hammerhal and I can imagine Josh doing a very delightful switch from sword and sorcery fiendishness into CJ Sansom-y, Age of Sigmar wonders. Untangling the jurisdiction of the gods, determining who a soul actually belongs to.
Outside Warhammer - did anyone ever read Aliette de Bodard's Aztec-noir detective trilogy - Obsidian & Blood?
Not only were they brilliant, they show a wonderful way of doing high-concept fantasy wonders right alongside serious history and pretty damn compelling crime fiction.
has even a hint of a similarity, we're in for a treat.