In some inferential problems involving Markov process data, the inhomogeneity of the process is of central interest. One example is of a binary time series of data on the presence or absence of a species at a particular site over time. Here the two states correspond to `presence' or `absence', respectively, of the species, and the main topic of interest is temporal variation in the process. In principle this variation can be modelled parametrically, but in the absence of information about the physical mechanism causing species numbers to fluctuate, it is usually very difficult to suggest a plausible model that explains the data, at least until a more adaptive analysis is conducted. These issues argue in favour of nonparametric methods for estimating probabilities of transition, and for estimating probabilities of the process being in a given state at a given time. Such techniques, which in practice might be a prelude to parametric modelling, will be introduced and explored, under assumptions motivated by characteristics of the data set mentioned above. These assumptions will be shown to lead to consistent estimation of probabilities, and so to imply that nonparametric methodology gives accurate information about properties of the process.