Let $P_1, \ldots, P_9$ be the following arrays of integers:
\begin{align*}
\left( \begin{matrix} 9 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & 7 \\ 6 & 8 & 9 \end{matrix} \right), 
\left( \begin{matrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 8 \\ 7 & 9 & 1 \end{matrix} \right),
\left( \begin{matrix} 2 & 3 & 4 \\ 4 & 5 & 6 \\ 6 & 7 & 9 \\ 8 & 1 & 2 \end{matrix} \right),
\left( \begin{matrix} 3 & 4 & 5 \\ 5 & 6 & 7 \\ 7 & 8 & 1 \\ 9 & 2 & 3 \end{matrix} \right),
\left( \begin{matrix} 4 & 5 & 6 \\ 6 & 7 & 8 \\ 8 & 9 & 2 \\ 1 & 3 & 4 \end{matrix} \right),
\left( \begin{matrix} 5 & 6 & 7 \\ 7 & 8 & 9 \\ 9 & 1 & 3 \\ 2 & 4 & 5 \end{matrix} \right),
\left( \begin{matrix} 6 & 7 & 8 \\ 8 & 9 & 1 \\ 1 & 2 & 4 \\ 3 & 5 & 6 \end{matrix} \right),
\left( \begin{matrix} 7 & 8 & 9 \\ 9 & 1 & 2 \\ 2 & 3 & 5 \\ 4 & 6 & 7 \end{matrix} \right),
\left( \begin{matrix} 8 & 9 & 1 \\ 1 & 2 & 3 \\ 3 & 4 & 6 \\ 5 & 7 & 8 \end{matrix} \right). 
\end{align*}
$P_i$ is obtained from $P_{i-1}$ by adding $1$ (mod $9$) to every entry of $P_{i-1}$.

These arrays of integers correspond to the following semistandard Young tableaux $T_1, \ldots, T_9$ respectively (these tableaux can be obtained from each other by promotion): 
\begin{align*}
& \left(\begin{array}{ccc} 1 & 2 & 4\\ 2 & 3 & 7\\ 4 & 5 & 8\\ 6 & 9 & 9 \end{array}\right),\left(\begin{array}{ccc} 1 & 1 & 3\\ 2 & 4 & 5\\ 3 & 6 & 8\\ 5 & 7 & 9 \end{array}\right),\left(\begin{array}{ccc} 1 & 2 & 4\\ 2 & 3 & 6\\ 4 & 5 & 7\\ 6 & 8 & 9 \end{array}\right),\left(\begin{array}{ccc} 1 & 3 & 5\\ 2 & 4 & 7\\ 3 & 6 & 8\\ 5 & 7 & 9 \end{array}\right),\left(\begin{array}{ccc} 1 & 2 & 4\\ 3 & 5 & 6\\ 4 & 7 & 8\\ 6 & 8 & 9 \end{array}\right), \\
& \left(\begin{array}{ccc} 1 & 3 & 5\\ 2 & 4 & 7\\ 5 & 6 & 8\\ 7 & 9 & 9 \end{array}\right),\left(\begin{array}{ccc} 1 & 1 & 4\\ 2 & 5 & 6\\ 3 & 7 & 8\\ 6 & 8 & 9 \end{array}\right),\left(\begin{array}{ccc} 1 & 2 & 5\\ 2 & 3 & 7\\ 4 & 6 & 8\\ 7 & 9 & 9 \end{array}\right),\left(\begin{array}{ccc} 1 & 1 & 3\\ 2 & 4 & 6\\ 3 & 7 & 8\\ 5 & 8 & 9 \end{array}\right).
\end{align*}

I think that there is a combinatorial rule to obtain $T_i$ from $P_i$. We can move elements which are not in the correct position to a correct position. It seems that it is similar to Jeu de taquin. But I have not figured out that. Any help would be greatly appreciated. Thank you very much.