Pseudotime algorithms can be employed to extract latent temporal information from cross-sectional data sets allowing dynamic biological processes to be studied in situations where the collection of time series data is challenging or prohibitive. Computational techniques have arisen from single-cell 'omics and cancer modelling where pseudotime can be used to learn about cellular differentiation or tumour progression. However, methods to date typically implicitly assume homogeneous genetic, phenotypic or environmental backgrounds, which becomes limiting as datasets grow in size and complexity. We describe a novel statistical framework that learns how pseudotime trajectories can be modulated through covariates that encode such factors. We apply this model to both single-cell and bulk gene expression data sets and show that the approach can recover known and novel covariate-pseudotime interaction effects. This hybrid regression-latent variable model framework extends pseudotemporal modelling from its most prevalent area of single cell genomics to wider applications.