A new class of time-dependent Dirichlet priors is introduced as a generalisation of the Wright-Fisher diffusion, allowing discontinuities in the trajectories, as well as non-Markovian memory. This class is obtained as a simple stochastic time-change (subordination), interpreted as a hyper-prior assigned to the operational time-clock of a Wright-Fisher diffusion. Explicit representations and exact sampling algorithms are obtained for prior and posterior distributions of the process and of its clock, given partially exchangeable data sampled at discrete time-points. Computability and conjugacy rely on a novel class of discrete dual processes, generalising existing results on duality and computable filters.