Running water is what’s currently recommend for first aid for superficial wounds. No soap, no hydrogen peroxide or betadine or iodine, all of which can cause cellular damage and slow wound healing. Any wound that’s deep enough, open enough or irregular enough that you can’t easily clean it under the tap needs professional evaluation and a prescribed treatment plan.
For longer term wound care, there are wound cleansing sprays and saline to clean and irrigate wounds. The one I use most (Sea-Clens) lists: water, decyl glucoside, sodium chloride (USP), citric acid, diazolindyl urea, methyl paraben, tetrasodium EDTA. It’s designed to soften and remove necrotic tissue. Again, no hydrogen peroxide or betadine or iodine.
I sometimes use iodoform (iodine reacted with ethanol or acetone and sodium hydroxide) tape to pack infected abscesses, but that’s because killing the infection and removing necrotic tissue is important, and iodoform doesn’t do as much damage to healthy tissues as elemental iodine. Still, once it appears the infection is gone, we usually stop the iodoform tape and switch to a plain gauze (or hydrogel) packing material.
The more we learn about wounds, the more we learn that our best treatment is often as little treatment as possible. Keep it clean, keep the wound bed moist and the skin around it dry, and don’t scratch!