Time to reach stationarity in the Bernoulli-Laplace diffusion model

Consider two urns, the left containing $n$ red balls, the right containing $n$ black balls. At each time a ball is chosen at random in each urn and the two balls are switched. We show it takes $\tfrac{1} {4}n\log n + cn$ switches to mix up the urns. The argument involves lifting the urn model to a random walk on the symmetric group and using the Fourier transform (which in turn involves the dual Hahn polynomials). The methods apply to other "nearest neighbor" walks on two-point homogeneous spaces.