The sculpture above is not from the Civil War. It's from 2001, depicting horses carrying Green Berets through the dangerous and mountainous terrain of northern Afghanistan.
Remembering fallen war heroes extends to the four-legged members of the armed forces who have sacrificed their lives over the years.
Explosive detection dogs have played a key role in the wars in Afghanistan and Iraq, saving many lives of many servicemen and women and innocent civilians.

At the Military Working Dog Teams National Monument website you will find the names of 25 K-9 handlers who have been killed in the lines of battle in Iraq and Afghanistan, as well as a history of battlefield canines from the creation of the working dogs program at the start of World War II through today.   

Some 230 war dogs were killed in Vietnam as did more than 150 dog handlers.
I just learned today that a bronze monument has been placed across from Ground Zero in New York City dedicated to the horses used in our most recent war.
These horses transported U.S. Special Operations forces - some with no riding experience - through Afghanistan after they parachuted into Taliban territory following 9/11. (Thank you @DrMartyBecker for this tip.)
The sculpture, designed by Douwe Blumberg and commissioned by an anonymous group of Wall Street bankers, includes a piece of steel from the fallen World Trade Center in its base. Titled "America’s Response Monument," the sculpture bears the subtitle, "De Oppresso Liber," Latin for "to liberate the oppressed" -- the motto of the Green Berets

Just a few miles down the road from me, under many inches of soil, no doubt still lie the bones of hundreds of horses that perished during the Battle of Gettysburg, which this year marks its 150th anniversary.