For many spatial scales and problems, geography can naturally be partitioned into a finite number of discrete sites fS k g for k~1, . . . , K. Examples of these situations include individual cities, islands or countries. Starting from the observed data, at the tips of the phylogeny F we record discretized locations X~(X 1 , . . . , X N ), where X i [fS k g pin-points the sampling site of taxon i. Unobserved in the spatial process are the locations of the most common recent ancestor X root drawing from root distribution p root , the times at which the descendent taxa move and amongst which discrete sites these moves occur, a process which ultimately gives rise to X. Conditioning on X root and the unobserved locations realized at each internal node (X Nz1 , . . . , X 2N{2 ) [13, 16] , suggest modeling the instantaneous locations X (t) for taxa along each branch in F as independent continuous-time Markov chains (CTMCs). CTMCs are the same processes one commonly exploits to model sequence character evolution [15, 50] . Although many readers are familiar with CTMCs, we here highlight several chain properties to which we turn later when discussing CTMC modeling limitations. CTMCs are the simplest stochastic processes that emit discrete outcomes as a continuous function of time. The processes are memoryless, in that the probability of transitioning to a new location only depends on the current location and not the past history. A K|K infinitesimal rate matrix L~fl jk g completely characterizes the CTMC process. Rate matrix L contains non-negative off-diagonal entries and all rows sum to 0, yielding a stochastic matrix upon exponentiation. Solving the Chapman-Kolmogorov equation that specifies the behavior of the chain yields the finite-time transition probabilities p jk (t)~Pr X (t)~S k j X (0)~S j À Á . In matrix form,