In 1910, Mr.and Mrs. D.H. Woolf walked from Kansas City* to NYC, with their dog. I'm unable at this time to tell you the name of their dog.
It's unlikely they were any more than the most distant relations of Virginia Woolf; and likely no relation of D.H. Lawrence at all.
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*glancing reference at the remarkable showhistory.com with a link to Con mezzi a traino animale at the equally remarkable La storia dei globetrotters.
In both instances the Woolfs are cited near the bottom of the page.
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"around the world within 5 years. These cards are sold to cover the expenses"
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And, by the by, Karl Bushby's still at it. His footprints are now in Russia, having begun their appearance in Tierra del Fuego and continued uninterrupted from there up the American continents and across the Bering Sea.