The website that Jay Dougal pointed us to has a number of cadences whose tunes and words appear to be modified versions of 19th/early 20th century African American dance songs and rhymes {that in the late 20th & early 21st century would be considered children's rhymes, but didn't necessarily originate as rhymes for children}.

For instance, see these running cadence chants with either the title of the title or the verse/s of the source songs or rhymes given in brackets:

"Sittin' on a mountain top, beating my drum
Beat it so hard that [t]he MP's come
I said MP, MP, don't arrest me
Arrest that leg behind the tree
He stole the whiskey, I stole the wine
All I ever do is double-time "

[...Oh, Mistah Washin'ton, Don't whoop me,
Whoop dat N**ger Back 'hind dat tree.
He stole tucky, I didn't steal none,
Go wuk him in de co'n field jes fer fun.]
-from T-U-Turkey in Thomas W. Talley: Negro Folk Rhymes, Wise And Otherwise {Kennikat Press Edition, 1968; pps 6-7; originally published, 1922

{also found as "policeman, policeman, don't arrest me" in some collections of children's rhymes}

**

Two old ladies lying in bed
One rolled over to the other and said,
"I wanna be an Airborne Ranger
Live a life of sex and danger
Blood, guts, sex, and danger
That's the life of an Airborne Ranger!"

[line #1-2; "Shortnin Bread"]

**

"AWOL, AWOL, where've you been?
Down in the bars, drinking gin
What ya gonna do when you get back?
Sweat it all out on the PT track"

["Hambone"]

**
There are other examples on that page, but here's a marching cadence posted there whose source song is very easy to identify:

"You get a line and I'll get a pole ) CC
Honey, honey ) G
You get a line and I'll get a pole ) CC
Baby, baby ) G
You get a line and I'll get a pole ) ALL
We'll go down to the fishin' hole ) ALL
Refrain: Honey, oh baby, be mine ) ALL
Go to your left, your right, your left ) ALL
Go to your left, your right, your left, hey! ) ALL
I had a girl who lived on a creek
Honey, honey
I had a girl who lived on a creek
Baby, baby
I had a girl who lived on a creek
She was cute and she was sweet
I had a girl, looked good in blue
Honey, honey
I had a girl, looked good in blue
Baby, baby
I had a girl, looked good in blue
She could make a fool out of you"

[The Crawdad Song]

Here are the instructions that were posted on that page for that song:

"This one doesn't follow the normal call and echo of most cadences. The cadence caller says the CC lines, and the group replies with the G lines, instead of echoing the CC lines, then all call the ALL lines, with no echoing."