Any hunts she would start would be earned by the hunted through misdeeds, like perma-killing Guardians or being involved in suspicious disappearances. Unlike what you think, Praxics are very by-the-book people, preferring having incriminating evidence first.

No lore that I've seen. She's probably just really suspicious of Stasis users, and is keeping an eye on some of them (which Zavala would be fine with). Since no Stasis users we know of have fully fallen into corruption, and/or caused a final death, my theory is that she is just snooping around, being really sus of Stasis users.