Suppose we have $n>1$ cells that are arranged in a row. Each cell contains a coin.
We label the coins uniformly by integers ranging from $1$ to $k$, where $k$ is chosen such that $\ell k = n$ for some parameter $\ell\geq 1$. The initial order of the labels is random. Then we move coins, and possibly stack them, according to the following process:
We roll a $k$-faced dice that shows number $1,2,\dots,k$ with probability $p_1,p_2,\dots,p_k$. All topmost coins labeled by the number shown on the dice are moved from their cell to the next cell (or from the last cell to the first cell). If there is already a coin in the cell, the coins are stacked. Only the topmost coin is moved.
To make things a little clearer, consider the following example: Let $n = 12$ and $\ell = 4$, so we have $k = 3$, i.e. each coin is labeled by numbers ranging from $1$ to $3$. There are three groups: four coins with a 1, four coins with a 2 and four coins with a 3. One possible order could be $$\begin{align*} \left(\begin{array}{c}1\\2\\1\\3\\1\\2\\3\\2\\2\\1\\3\\3\end{array}\right)\end{align*},$$ i.e. coins 1, 3, 5 and 10 are labeled by $1$, coin 2, 6, 8 and 9 by $2$ and the remaining coins by $3$. If we now roll a dice and the face shows $1$, we would move coins 1, 3, 5 and 10 to their next cells and continue with the next roll.
What is the name of this stochastic process if we are interested in the number of coins at round $t$ in a cell? Is this game a known and studied game stochastic process?
The Totally Asymmetric Simple Exclusion Process. Particles try to hop from point to point clockwise around the ring at random (exponentially distributed) intervals. In the single-class case, illustrated by Fig. 1, the only additional complication is that the progress of one particle may be blocked by another occupying the site ahead. The small arrows in the figure indicate possible hops, those that are not blocked. Any such transition will occur in the next instant $dt$ with probability $dt$.
We can use this model to address your problem specialised to the case $\ell=1$ and $p_{i}=1/k$ for all $i$. Roll the die and one of the coins is selected at random. If it is the topmost coin in its cell, it has a cell barrier immediately to its right. It will hop over that barrier into the next cell leaving the barrier to its left, just as a particle would hop into the hole to its right leaving a hole where it was before.
This is not as general as what you were asking, but it is already a very interesting process, tractable to analysis yet exhibiting phenomena comparable to shock waves and condensation. Applications include lattice gases and traffic jams. This article on the zero-range process could be a start. There is also a version with multiple classes of particle but I don't think it's the generalisation you want.