Convergence of some time inhomogeneous Markov chains via spectral techniques

We consider the problem of giving explicit spectral bounds for time inhomogeneous Markov chains on a finite state space. We give bounds that apply when there exists a probability π$\pi$ such that each of the different steps corresponds to a nice ergodic Markov kernel with stationary measure π$\pi$. For instance, our results provide sharp bounds for models such as semi-random transpositions and semi-random insertions (in these cases π$\pi$ is the uniform probability on the symmetric group).