I really wouldn't use the dishwasher for anything that hasn't already been washed out, as far as anything that has been scented. Reason being, you never know how AWFUL something smells when it is in a hot & humid environment. Yes, my family member stuck my glass measuring cup with Dragon's Blood that I hadn't washed out yet straight into the dishwasher as I had to pour into the mold and run to town for dinner - she loaded the dishwasher while I was gone. Took 2 months (even with vinegar) for most of the smell to dissipate. Even now 9 months later, I can once in awhile get a faint whiff of DB. And the hot & humid smell of it just turned me off of it and makes me sick to my stomach to smell DB now. So now I can't even bring myself to make more DB soap even though my best friend loves it.
If I'm doing CP, I mix up and pour into the mold, scrape as best I can, take a couple of paper towels or pages of newspaper to swipe at the worst of it, then set the stuff in the garage out of the way until the next day or the day after. By then, its generally saponified so I just wash it by hand. Nice to not have to actually use any dish detergent to wash them up. I do still tend to use gloves because the soap is still harsh compared to what it will be in a few weeks. With HP, I just pour into my mold, stick the crock in the sink and let the water run hot, then fill the crock and let it soak while I'm banging the air out of my soap and putting plastic wrap on top.
I have a dedicated pitcher that I use to mix my lye solution in - I just let it sit in the sink and run over with water for a minute or two while I'm stickblending then I give it a good full dump and fill two or three times before setting it out to dry after I've poured into the mold (or while my HP is cooking). I don't like to wash my lye pitcher with soap because it usually causes floaters for me (my pitcher is PP5 plastic) and I absolutely hate having to find my strainer for my lye.