Most of the time I use shampoo (provided by many hotels) to wash clothes. However, when traveling in Africa, I carry bars of detergent made specifically for hand washing. On my recent trip to China, I bought two bars of Tide detergent (Procter & Gamble) in Beijing. Sunlight (Unilever) is available in India and Sri Lanka. There are many local brands as well.
Carry-on traveling pack (size?)
We just visit a local store when we land and buy a small amount of detergent for our washing. We usually don't pack shampoo, toothpaste, deodorant, shaving cream, etc. to save luggage space when we're going for a month. We buy them when we land and use them up while we're there. It only takes a half hour to get all this stuff and it's fun to look through the store and see what's available. Everyone always wants to "meet the locals" and this is one good way to do it.
Anyone else read this title and think it was an old movie title?
===Carry On Travelling: (pack size)===
No? Just me? ah well.
[ Edit: Edited on 23-Nov-2018, at 14:02 by Piecar ]
Yes Yes and Yes. I've done many times when travelling with hand luge only. Liquids are the most issue, but then you pick up them locally in small sizes that you do use for few days and leave them behind. They are cheap anyway. Depends on the weather, but you always can pick up few extra cheap T-Shirts for €5 or so along the way. 
Now, you say you know your airlines' rules, but I can hardly imagine that the smaller budget airlines in Europe will handle your 55L pack as carry-on. Just to help you out.;
It seems that 40L usually is the largest also to be allowed at any time on most airlines world wide, including most budget airlines in Europe and Asia, where limits are sometimes more strictly enforced.
Enjoy!
I recommend the Mountain Hardwear Rainshadow 36. $170 on Amazon. I've switched to 100% quick dry synthetic clothes. Period. (Try carrying a pack of dirty (heavy) cotton fabric. No fun.). I also recommend a very small/light day pack like the Osprey Ultralight. In this I carry absolute essentials (phone, passport, water bottle). Sometimes (like bus from Costa Rica to Nicaragua), even the 36L pack was required to be stored under the bus.
Post 17 was removed by a moderator
lIn the past (before there were wheeled bags) I went all over Europe with one 20" Samsonite suitcase and a large purse. I am sure that I could fit everything that I had on that trip into a carry-on sized bag and probably had room left over. I carried powdered Woolite for laundry. More recently (1998) I had two carry-on sized bags for a trip to Belize - in those days I did not have a computer with me, but I did have my diving gear and a lot of travel information.
After 2006, I packed all my 'stuff' in a small wheeled bag which met the requirements for carry-on size for most of the major airlines. But I usually checked the bag because I didn't want to be bothered with shlepping it through the airport. I'd rather wait for it at the end. And of course now I have my mobility scooter to deal with. I no longer have the energy to shop for stuff at my destination, and I don't know if I ever did, except for food for breakfast and lunch in the room.
I don't bother with jewelry or scarves to accessorize. I take two pair of very comfortable shoes (plus the ones I'm wearing), a hat with a brim and a chin strap (so that it doesn't blow off), a short raincoat, a fleece, one or more bathing suits, and several tops, and skirts and a few dresses. I don't like wearing trousers or jeans, and I never wear shorts.
We each have a carry-on bag that we check through. Then we carry on a school-sized backpack. This serves for a month or longer. It would serve for a year if we were gone that long. You need a week of clothes and enough layers to cover weather changes. That's all. Anything else is stuff you have to carry around and will never use. If I did't take my computer and painting supplies, I wouldn't need the backpack . . . except I do like to bring home gifts for people. It's handy for that.