Let $P_1,...,P_k$ denote the transition matrices of $k$ irreducible, aperiodic Markov Chains on finite state spaces $S_1,...,S_k$ all of them being disjoint. Hence, there are stationary distributions $\pi_1,...,\pi_k$ such that $\nu_iP_i^k\to \pi_i$ for any initial distribution $\nu_i$. Denote the corresponding mixing times by $t_i$.
Define by $P=\mathrm{diag}(P_1,...,P_k)$ a stochastic block diagonal matrix. Then $P$ can be seen as a transition matrix for a Markov chain on the state space $S=S_1\times...\times S_k$ and converges for a given initial distribution $\nu$ on $S$ to $\pi = (\nu(S_1)\pi_1,...,\nu(S_k)\pi_k)$. I am wondering what I can say about the mixing time $t$ of the chain $P$. Intuitively it should be something like the $\max\{t_1,...,t_k\}$ since "the slowest chain" amongst the $P_i$ should dominate the convergence speed of $P$ but I am not really making any progress on showing this.
Maybe my claim is even wrong. Any help is greatly appreciated.
EDIT: Definition mixing time:
$t_m=\inf\left\{t\in\mathbb{N}:\max_{x\in S}||P^t(x,\cdot) - \pi||_{TV}<\dfrac{1}{2}\right\}$.