Green’s Dictionary of Slang

to amble, to trudge along slowly.

1772		Bridges Burlesque Homer (3rd edn) 62: With joyous phizz away they mog off.
1797		Bridges Burlesque Homer (4th edn) II 128: The rest, I hope, will scorn to mog off. / And dim my day-lights if I jog off.
1895		DN I 398: Mog [...] to walk.
1927		‘Expressions of the Maine Coast’ AS III:2 139: Mog meaning to move slowly, to depart.
1938		B.C. Damon Grandma Called It Carnal 262: They mogged slowly all the way home in a delicate silence [OED].
1950		in Time 22 May 24: Exhausted federal mediators who dutifully mogged back and forth trying to find a formula [HDAS].
1989		H.F. Mosher Stranger in the Kingdom 16: Val [...] rolled her eyes toward the ceiling and mogged back out to the kitchen [DARE].
1993		in DARE file.