<p>Microflows like Knudsen pumps often include rarefied gases featuring different degrees of rarefaction. This different modeling complexity requires space- and time-adaptive rarefied gas models that resolve the physical effects efficiently in each subdomain of the microflow. Different-order moment models are effective at describing rarefied gases in microflows with respective levels of complexity in each subdomain. In this work, analytical model-error estimators for the Hyperbolic Moment Equations (HME) model are derived that are then used to define a space- and time-adaptive moment model for microflows of rarefied gases. First, a domain decomposition strategy of the microflow into subdomains, each modelled by an HME model of an appropriate order, is presented, using domain decomposition criteria that are based on the exact model difference between a higher-order HME model and a lower-order HME model and on chosen error thresholds. Secondly, a non-linear adaptation of a recently developed padded buffer cell approach is presented to couple these varying-order HME models using a single finite volume scheme. Finally, a smoothing of the domain decomposition is proposed to limit oscillations generated by the coupling. While the performance depends on the thresholds and the smoothing parameter, the proposed adaptive moment model is able to capture the different degrees of rarefaction of the rarefied gas in the microflow and yields accurate numerical results compared to experimentally validated DVM benchmark data while obtaining computational speedups of up to 40 percent compared to using a high-order HME model in the entire domain.</p>