I would imagine there are as many ways that it could be done as there are folks doing it, but here is the way I do it and why.
Few things can wreak more havoc on the inside of steel than forging, bladesmiths have a lot of odd extra steps we will go through to make the steel "better", but what we are making it better from is the forging. From the mill it is pretty good, then we proceed to beat the snot out of it with very uneven methods, while heating and cooling it in very uneven ways. So needless to say I like to do my normalizing right after the forging.
Industry suggests normalizing at temps much higher than most bladesmiths prefer, after the first sparkler show from a stub that was a half finished blade, smiths begin to fear highers heats. But those heats used in industry have their benefits so I incorporate them into what I do.
So my first normalizing heat will be at around 1600F to get everything into solution in order to even things out. Yes I may get larger grain at this step but I don't care what size the grains are as long as they are all the same size. The subsequent normalizations will be at cooler heats in order to refine things. The next will be around 1500F. and the final heat at or slightly below critical.
The operative word in normalizing is "even". Heating as evenly as possible and cooling as evenly as possible is essential. Carefully heat until the entire blade is all the same color at the same time and then put the tip of the tang in a vice and let the things air cool as evenly as possible. Tossing it on the floor or the anvil is a no-no. The wood ash trick is annealing, not normalizing, the speed at which it cools is irrelevant as long as it is even.