From what I read in the fluff the Adeptus Arbites should be in charge of all crimes involving any of the Adeptus Terra. But is this really true? Any of them? Lets say a group of Arbitrators catch a renegade Tech-Adept would they just execute him on sight or rather hand the Adept over to the AdMech who after some questioning will probably punish her/him by repurposing as a Servitor.
Same goes for the Ecclesiarchy: Would the Adeptus Arbites directly judge a rogue priest or would she/he rather be transferred to the nearest Church for internal judgement?