The Frank C Brown Collection of North Carolina Folklore lists a variant of the lyrics Joe posted, but not much in the way of explanation except that it was described as a lullaby by the source of this version.

The old grey horse came trotting down the wilderness
Trotting down the wilderness
Trotting down the wilderness
The old grey horse came trotting down the wilderness
Down in Alabam.

Roll, Riley, roll
Roll, Riley, roll
Roll, Riley, roll
Oh, Lord, I'm bound to go

VAnce Randolph's "Ozark Folksongs" contains some other verses:

Little black bull come down in the wilderness....
Long time ago.

Was an old mule and he come from Jerusalem..
Down in Alabam

Barnum caught him, put him in his museum....

Ain't you mighty glad to get out o' the wilderness...

Fifteen cents to get out o' the wilderness....

Randolph also indicated that a related religious marching song called "Go in the Wilderness" appeared in the Atlantic Monthly in 1867.

Then there is the version that goes :

The old grey mare went Ptthhft! down the Delaware ...
Many long years ago.

The old grey mare went Ptthhft! on the whiffletree...
(The Ptthhft! is a sound commonly called a "raspberry")

Parodies from WWI era:

1. The poor old Kaiser ain't what he used to be... Many long years ago.

2.Oh we don't have to hike like the infantry,
Ride like the calvary
Shoot like the artillery
Oh, we don't have to fly over Germany
We're the Q.M.C.

We're in the Q.M. C.
We're in the Q.M.C.
Oh we don't have to hike......

3. Oh Uncle Sammy, he pays the infantry
He pays the calvary,
He pays the artillery,
And then, by gosh, he closes the treasury
To hell with the engineers.

4. Oh, Uncle Sammy, he needs the signal corps,
Wigwag and semaphore
He gets them by the score,
And then, by jing, we're off for a foreign shore,
Good-by, Kaiser Bill!

From the University of Minnesota:

I don't give a damn for the whole state of Iowa....
'Cuz who the hell likes corn.

Finally (for now) there is the scholastic medley of "Row your boat" and "Old Grey Mare"

Row, row, row your boat
Gently down the stream
Throw the teacher overboard
And listen to her scream.

Five days later she couldn't find her underwear
Couldn't find her underwear,
Didn't have a thing to wear*
Ten days later she was eaten by a polar bear
And that was the end of her

*could this refer to a blue clicky thing that she was lacking?

rich r