I am not sure what the process and purposes are for other smith interrupted techniques, but here is what the concept is from a metallurgical point of view and why I would do it.
Austenite is FCC (face centered cubic) which is a most efficient atomic stacking arrangement. All the other transformation products that occur on cooling are BCC (body centered cubic)or body centered in nature like the BCT(body centered tetragonal) of martenensite, which takes much more space for the same amount of atoms. So while we do get simple thermal contraction as metal is cooled it is fairly small compared to the massive expansion you get when you have a BCC transformation. This all seems academic, but some folks may not realize this. I recently read an article that stated that steel contracts when martensite forms, this is so completely opposite of the facts that I was shocked the editors let it get by them, it is quite basic and fundamental that the martensite transformation is one of the most expansive.
So when you cool a blade and one part cools quicker than the other it will reach a body centered (hopefully martensite) transformation sooner. If this happens you could have expansion on one part of the blade while another is still contracting, it is obvious what the dangers of this may be, and since the part that has not yet transformed is still very plastic austenite, there is plenty of room for permanent shape change. If you interrupt the quench just before this transformation and allow the entire blade to equalize in temperature before continuing, it becomes very obvious how advantageous it can be to have the entire blade hardening evenly at the same time.
It is best done as near the transformation temperature as possible without beginning it. In oil I like to pull the blade out at the point where there is just a few wisps of vapor coming off the steel and the oil is able to coat the blade without burning off or flashing. Many oils have a vapor point that approximates 400F. so that can be helpful.
I have hade better luck with clay in oil by having a soft wooden block to tap the spine of the blade on and knock the clay off before allowing it to cool. When you are playing with clay you are dealing with a more complicated process, since the pearlite transformation will also be BCC and will have to be dealt with as you are intentionally making pearlite as well. The idea here is the same but a little trickier since you need to balance the martensite formation with the pearlite formation. Have you ever heard about folks complaining of a reverse curve when the quenched a clayed blade in oil? The pearlite formation got away from them and the tip dropped. I believe this phenomenon is very much more complicated but I need to have data to back my theories before I am comfortable sharing them. Anyhow the idea behind interrupting the quench in water or brine is to slow things down enough to even out the transformations before martensite runs away with the process and you hear that awful ping.
Is anyone else's head spinning? 
This is great info but I had to save it and set it aside so I can wait and wrap my brain around it when it's ready. 
Long week, you know...
this cause is mentioned in many a thread 
