Biologically there is no difference between an Exodite, Corsair, Craftworld, Harlequin, or True Kin. They are just different cultures and in the case of True Kin they have allowed their Psionic abilities atrophy so as to better protect themselves from the beings in the warp while the other cultures have not taken this approach.
Like you said Craftworlders go to their spirit stones, as Betray said True Kin, if they can afford it, regenerate their bodies and their souls are some how bound to their cells. Exodites have spirit stones I would assume as well as they have infinity circuits on their worlds, and Corsairs are drawn from all groups save the Harlequins, so some are from the Exodite worlds, others are craftworlders, and some are True Kin. I haven't come across anywhere that specifically states what happens to their souls. Harlequins are supposedly protected by the Laughing God.
What do the True Kin do with their cousins when they take them in battle or a raid. Well truthfully I would say they do what ever they feel like. But very few of them would ever think about empowering Slaanesh. It is mentioned in the 5th edition fluff that the incubi take spirit stones and twist them making them into a weapon. Psykers would be handed over to the Haemonculus to create Crucibles of Malediction most likely. Maybe some Archons might even attempt to convert the occasional cousin by exposing them to the exhilarations of their own culture and get them to become pain junkies too. I really think it depends on who the Archon is and what their particular personality quirks would have them doing to prisoners of any race.
If taken alive they would need to suppress their Psionic abilities to make sure they don't break the taboo in the city and bring the wrath of Vect and the other Kabals down on them.