In the limit $t\rightarrow \infty$ we require that the two distributions $w_i(t+1)=w_i$ and $w_i(t)=w_i$ and we have $$\begin{equation*} \sum_j w_jT_{j\rightarrow i} A_{j\rightarrow i}= \sum_j w_iT_{i\rightarrow j}A_{i\rightarrow j}, \end{equation*}$$ which is the condition for balance when the most likely state (or steady state) has been reached. We see also that the right-hand side can be rewritten as $$\begin{equation*} \sum_j w_iT_{i\rightarrow j}A_{i\rightarrow j}= \sum_j w_iW_{i\rightarrow j}, \end{equation*}$$ and using the property that $\sum_j W_{i\rightarrow j}=1$, we can rewrite our equation as $$\begin{equation*} w_i= \sum_j w_jT_{j\rightarrow i} A_{j\rightarrow i}= \sum_j w_j W_{j\rightarrow i}, \end{equation*}$$ which is nothing but the standard equation for a Markov chain when the steady state has been reached.