 Random adjacent transposition shuffle

On the symmetric group $S_n$, this Markov chain chooses uniformly from the identity and the $n-1$ adjacent transpositions, and multiplies on the right. Its stationary distribution is the uniform distribution on a finite set, because the generators are self-inverse. A fixed label's position is a lumped Markov chain on a path graph with off-diagonal transition probabilities $1/n$ to existing neighbours.