Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the five stones are positions $(-2,-1,0,1,2)$.

I then proceed to do the following:

For each of some arbitrary number of discrete time steps, I select one of the $k$ stones with uniform probability. I then lift the stone up, uniformly select an unoccupied site at most a distance $d$ from the stone's original position (including the original position), and place the stone back down at this selected site. If $d=1$, no stone can be moved from its original position and the center of mass of the stones/system, $C_m$, will be immobile. However, if $d=2$, starting from the state $(-2,-1,0,1,2)$, the center of mass will move to either the right or left with some per step probability of $\frac{1}{k}$ (until we leave this initial system state).

My question is - provided some number of stones $k$, and some "leashing distance" $d$, how can we characterize the dynamics of the random walk taken by the above system's center of mass, $C_m$? What is the average step time and size, and, in terms of Euclidean distance, what mean square displacement can we expect after some number of steps, $T$?