All of the above. Some of it gets bucketed onto the compost heaps, but most gets drained onto the paddock. We have acidic soils here so the high pH is a bonus for watering, but I dilute it about 25% so that roots don't get a shock.
I quench it twice: Once to put out the fire, and a second time when I put it in a ceramic tub to cool it off. I don't want it to smolder for weeks. The first, fire quenching water just drains off. With the second, ceramic tub water, I pour it off onto alkaline-tolerating plants before crushing the biochar.