This won't help you guys with bushel-size ash tubs, but here's what works for my little Franco-Belge hopper-feed stove.
The F-B uses an ash pan about 20" by 9" by 5" deep. I have two of these pans. I shake the grates every 8 hours. One full day of ashes fills one ash pan, which I dump outside late at night before the pre-bedtime shaking.
For dumping the pans, I use two small round metal ash cans with a lid and handle, each of which holds about 4 days of ashes.
Before my bedtime ash shaking, I take that day's full pan out of the stove and put an empty pan in. I set the full pan aside and cover it with a piece of heavy-duty aluminum foil that I've folded an edge into, so it fits tightly over the pan. This keeps any airflow in the basement from picking up fly ash.
The next night, before the last ash shaking, I take the full pan from the night before out to my back stoop, where I keep the cans, and dump the pan into the can. Since I only dump cooled ashes, I keep the ash cans lined with heavy-duty Glad or Hefty stress-flex bags with the built-in handle ties.
Like other posters, I've found there's not much you can do about the ash--no matter how gently you dump the pan, an ash cloud flies up. My pans don't weigh that much, so I can dump them with one hand and sort of just stand out of the way, and upwind.
The night before trash day, I pull the bags out of the cans and tie them shut. This is where spending a few cents extra on the stress-flex bags pays off, because the metal cans have the two sharp ends of the can's handle projecting a bit inside the can. The stress flex bags will give a bit and not tear open if they catch on anything while removing them. Nonetheless, caution is needed here.
Then I just put the two bags out with the trash. I've done it this way for about ten years now, and an added benefit is that the trash guys don't have to dump two full ash cans into the back of their truck, as they did before I started using the bags. That couldn't have been pleasant.
Whoever posted the pic of the bushel-size ash tub brought back some memories. I remember my Dad using those in the early 60s in our house just outside Bloomsburg, PA. I think nearly every house in town had a coal furnace at that time. In 1963 we moved into another house in Bloomsburg, which had a coal stoker with a long, enclosed auger to feed the coal from the coal room next to the furnace room. It really was a room, too--based in my 5-ton coal bin, I'd guess that coal room would have held 15-20 tons easily. I seem to recall the cement floor of the room was sloped to the center, in order to gravity-feed the coal to the auger.
Not sure how common this was, but that furnace also had an ash auger, which ran to a lid that fitted a standard size metal ash can, and automatically dumped the ashes into the pan. Every day or so, Dad would switch the auger lid to an empty can, and set the full can aside till ash day. Like many coal region towns, Bloomsburg had a separate ash pickup day, and if I recall correctly, they saved at least some of the collected ashes to spread on the roads during the winter. We didn't watch for salt trucks on a snow day--we looked for cinder trucks.