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Gaussian processes are valuable tools for non-parametric modelling, where typically anassumption of stationarity is employed. While removing this assumption can improve pre-diction, fitting such models is challenging. In this work, hierarchical models are constructedbased on Gaussian Markov random fields with stochastic spatially varying parameters. Im-portantly, this allows for non-stationarity while also addressing the computational burdenthrough a sparse banded representation of the precision matrix. In this setting, efficientMarkov chain Monte Carlo (MCMC) sampling is challenging due to the strong coupling aposteriori of the parameters and hyperparameters. We develop and compare three adaptiveMCMC schemes and make use of banded matrix operations for faster inference. Furthermore,a novel extension to multi-dimensional settings is proposed through an additive structure thatretains the flexibility and scalability of the model, while also inheriting interpretability fromthe additive approach. A thorough assessment of the efficiency and accuracy of the methods innonstationary settings is presented for both simulated experiments and a computer emulationproblem.