“Shit rolls downhill”

“Shit flows/rolls/runs downhill” means that difficulties flow from the top of an organization to the bottom, and those on the bottom will get the “shit.” The saying has been used in the military and in hospitals since at least 1970.
 
“Shit rolls down hill” was cited in print since 1970 and “Shit runs downhill” from 1971. The original proverbial saying is that “water runs downhill.”
       
 
Google Books
The Third Face of War
By Gene Schulze
Austin, TX: Pemberton Press
1970
Pg. 274:
“If he wanted to go from top . . . echelon . . . trust, travel, down to dealing with the enemy- Shit rolls down hill.”
 
Google Books
Time’s Children:
Impressions of Youth
By Thomas J. Cottle
Boston, MA: Little, Brown
1971
Pg. 280:
“It all goes downhill, right? Shit runs downhill. It’s all bullshit, right? It’s none of my goddamn business.”
     
Google Books
Hospital Chaplain
By Kenneth R. Mitchell
Philadelphia, PA: Westminster Press
1972
Pg. 61:
The standard saying in surgery at MUH was: “The shit flows down from the top.”
 
Google Books
The Choirboys
By Joseph Wambaugh
New York, NY: Delacorte Press
1975
Pg. 126:
He said, “I know shit rolls downhill. But why am I always livin in the valley?”
 
Google Books
The Unkindest Cut:
Life in the Backrooms of Medicine
By Marcia Millman
New York, NY: Morrow
1977, ©1976
Pg. 65:
As one surgeon pointed out, there is a widely used expression in his field: “In surgery the shit rolls downhill.” A cardiac surgeon explained that the different ranks of the house officers in surgery had to be recognized carefully: ...
 
Google Books
Maledicta
Volume 4
1980
Pg. 168:
From his point of view the infantryman may well consider himself lower than shit— he catches it from the enemy in combat and from his own Army in garrison, proving the Old Army adage that “shit rolls downhill.”
     
Google Books
Mother Jones
Volume 5
1980
Pg. 38:
The other, by a professor of medicine three years later to an uppity student: “Shit flows downhill and don’t forget that you’re on the bottom.”
 
Urban Dictionary
shit rolls down hill
when you have to deal with a problem that someone above you (your boss, parents, teachers, etc) caused.
don’t forget….you can throw shit up hill cause it will roll right back down, but heavier.
I gotta do your job cause you fucked up?
Man, shit keeps rolling down hill at this crappy job.
by metamorphosis May 15, 2004
 
Google Books
Sh*t Runs Downhill
By Kevin W. Batch
Houston, TX: Strategic Book Publishing and Rights Co.
2012