Two-coordinate stochastic averaging ($\left(\begin{array}{cc}t& 1-t\\ 1-t& t\end{array}\right),0\le t\le 1$)

This doubly stochastic matrix replaces two values $a,b$ by $ta+(1-t)b$ and $(1-t)a+tb$, preserving their sum. It fixes all other coordinates when embedded in a larger matrix. A finite product of such averaging matrices and permutation matrices is again doubly stochastic.