An arson rather than a looting example: In June and July of 1864, a Confederate Army (the "Army of the Valley") demanded enormous cash ransoms from several towns in Maryland and Pennsylvania. Hagerstown and Frederick paid up. Chambersburg was unable to raise the required sum and was burned to the ground on the orders of Gens. Jubal Early and John McCausland. See https://www.google.com/amp/www.baltimoresun.com/ph-ce-eagle-archive-0710-20110706-9-story,amp.html and https://en.m.wikipedia.org/wiki/Chambersburg,_Pennsylvania#/search .
The money was supposedly for the Confederate war effort, but it's hard to imagine, knowing Jubal Early's character, if some of it didn't end up in his pockets, just like the reward for the enemy commander being asked about.
After spending some years in self-imposed exile after the war, Early and McCausland were pardoned by Presidents Johnson and Grant, respectively.
Gen. Joseph Johnson refused Gen. Early's orders to similarly burn Hancock and Cumberland to the ground for not paying their ransoms. So even in 1864 some people recognized that you couldn't claim "just following orders" as an excuse for committing a war crime....