Please remember this; "you do absolutely everything with your hands", so, I use latex gloves all the time even going to parks, it seems ridiculous but, we do change many times and different children. They're not that expensive and are tax deductible.
I keep all or at least do all the diaper changes in the bathroom (hygienic reasons)
Each child has his/hers own changing pad. You can also labeL each, I try to avoid cross contamination. When I go to parks I have only one pad but I carry a few news paper pages in my change bag I just throw in the bag with a dirty diaper that way the mat still clean for the next change. I use also gloves and I carry soapy water and disinfectant. Along with my first aid kit.
It just made sense to keep a cosy common changing pad, heavy duty cleaning paper, a box of gloves and disinfectant, in the bathroom it makes really easy my work.
After a change it makes easy to spray and wipe wash his/hers hands and my hands and put all things away. If each child has their own pad makes it easy when one gets too messy you can wash or deal with that later on and not to worry about the other children in need of change, you just go ahead with their own pads. And last deal with the messy one.