  We consider a generalization of the Bernoulli-Laplace model in which there
are two urns and $n$ total balls, of which $r$ are red and $n - r$ white, and
where the left urn holds $m$ balls. At each time increment, $k$ balls are
chosen uniformly at random from each urn and then swapped. This system can be
used to model phenomena such as gas particle interchange between containers or
card shuffling. Under a reasonable set of assumptions, we bound the mixing time
of the resulting Markov chain asymptotically in $n$ with cutoff at $\log{n}$
and constant window. Among other techniques, we employ the spectral analysis of
arXiv:0906.4242 on the Markov transition kernel and the chain coupling tools of
arXiv:2203.08647 and arXiv:1606.01437.
