  We consider the problem of defining and fitting models of autoregressive time
series of probability distributions on a compact interval of $\mathbb{R}$. An
order-$1$ autoregressive model in this context is to be understood as a Markov
chain, where one specifies a certain structure (regression) for the one-step
conditional Fr\'echet mean with respect to a natural probability metric. We
construct and explore different models based on iterated random function
systems of optimal transport maps. While the properties and interpretation of
these models depend on how they relate to the iterated transport system, they
can all be analyzed theoretically in a unified way. We present such a
theoretical analysis, including convergence rates, and illustrate our
methodology using real and simulated data. Our approach generalises or extends
certain existing models of transportation-based regression and autoregression,
and in doing so also provides some additional insights on existing models.
