Quickly draw a downed animal before continuing the hunt. Keep animals clean as you transport them from the field. This is more of a problem with large animals, such as deer, that you may have to drag from the field. Wrapping the carcass in a plastic tarp or placing the carcass on a plastic sled will keep it clean and make it easier to drag. It also protects the hair on the hide, preserving the trophy value.
Temperature is another consideration.
When hunting in warm weather, be prepared to keep the carcass cool. You can place small carcasses in an ice chest. But, with deer killed when temperatures are above 50 degrees, transport the carcass to a walk-in cooler or place it on ice. At temperatures of 40-50 degrees, hang the carcass in a shady spot. Skin it and prop it open to promote cooling.