<b>Example</b>. $k=5$, $d=3$, $100$ (substantive) iterations; center of mass pink, time growing downward:
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<img src="https://i.sstatic.net/Q71j4.jpg" alt="InchWorm100" />
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It appears the center of mass performs a scaled random walk, scaled as a function of $k$ and $d$.
Perhaps scaled by about $(d/2)/k$...?