For those who don't take multi-week trips, who don't have an extended period between laundry days or who don't have the laundry of multiple persons to contend with, the bucket arrangement, although a bit messy at times, works fine.
I think Jana hit on something with the bucket/mop wringer combo. If you are washing 2 sheets, underwear, a few t-shirts, a pillow case and a couple of pair of pants it will take multiple bucket loads but its doable. I know because I did my wash with a toilet plunger and a huge paint
bucket when my washing machine broke down. (I also found out it was more fun to put everything in the bathtub, turn on an old Janet Jackson recording and dance everything clean.)
The wringing of larger items and bucket refills are the most cumbersome aspect of the project. Mop bucket wringers can be plastic, detachable and pretty light weight
. One could be stuck in the tow vehicle or stuck out of the way. A good bucket with a lid can also be a campers friend for other uses.
Washing powder tabs are wonderful for traveling. I used to buy the trial packages of powder for business trips that were longer than a week. At that point, I'd find a laundromat. The single-use trial packs were small and much cheaper than the single packs for sale
in the vending machines at laundromats. The tab packs are even more compact.
I think that on a trip long enough to require the doing of laundry, I'd like the bucket back up. Not all laundromats one stumbles to along the way are great places for a female to wander into solo.