"Dot and carry one" is found in the poem "Gunga Din" by Rudyard Kipling. Gunga Din is an Indian "Bisti", a servant who carried water for the troops, Kipling writes, referring to the Bisti carrying a water flask during a battle, "He would dot and carry one 'til the longest day was done, and he didn't seem to know the use of fear." I presume it meant that he scrambled to keep up with the troops.
Jackaline Winspear (in the novel "Birds of a Feather," uses the term to describe her assistant, whose leg had been seriously wounded in WW I, coming up a flight of stairs. This would seem to mean that his step was irregular because he had to favor his wounded leg.
Allen Peacock, April 8, 2019.