If Balefire doesn't destroy the soul, why then can't the DO lay claim to it? He is outside of space/time as he is not part of the Pattern so the burning a few seconds/minutes away shouldn't be a problem for him... Seems to me that the soul is simply burned from the Pattern too, and thus no longer accessible to ST.

Robert Jordan Answers: There are definitely time constraints on the Dark One's power to transmigrate a soul. The soul doesn't have to be secured immediately - that is, the Dark One doesn't have to be ready to snatch the soul at the instant of death - but the longer that passes after the death, the less chance that the Dark One will be able to secure the soul. Someone who has been killed with balefire in actuality died before the apparent time of his or her death, and thus the window of opportunity for the Dark One to secure that soul for transmigration is gone before the Dark One can know that the soul must be secured unless the amount of balefire used is very small. Remember that the more balefire is used, the further back the target's thread is burned out of the pattern.

After the soul is secured, then a suitable body must be acquired and stripped of the (former) owner's memory and soul to make way for the favored one. By the way, what constitutes a suitable body from the Dark One's perspective is not that of the recipient. Certainly Aginor would never have chosen to be reincarnated in his, shall we say, less than imposing body, nor would the womanizing Balthamel have chosen to be reincarnated as a beautiful woman. It was only chance that Moridin ended up in a body that is young, fairly good looking and physically imposing. Those things simply don't matter to the Dark One. But the body has to be basically healthy and sound, and neither too young nor too old. After all, the Dark One wants his servants to be effective, and a body that meets those basic requirements is more desirable than one that doesn't. Since there is no stockpile of such bodies, the only way for someone to die and immediately be reincarnated would be a matter of pure chance. That is, the death occurred when a suitable body was on hand for some other reason.

There are a few other limits and constraints, but I won't go into them here, since I may want to use them in the books, and I would rather they come as a surprise if I do.