What happened to the corpses of guys killed in running gunfights in the Old West?
If possible, those close to the deceased would bury the bodies in a shallow grave. If they were burying a friend or ally, they would cover the grave with heavy rocks, so wild animals wouldn’t come along and dig up the remains.
If the grave site was in hostile Indian country, they would camouflage the grave as best they could, so the bodies would not be disturbed.
If they were in a desperate situation and had to save their own hides, they had no choice but to leave the bodies where they fell.
When gunfighters bit the dust, anything valuable on the persons was taken off the bodies. If nobody had time to check, sometimes the killers just left the valuables where they lay. Several years ago, I found one such valuable: a rusted, old conversion pistol on the Santa Fe Trail in New Mexico.