Moment equations are a promising approach to model gas flows at microscopic scales, where the classical Navier-Stokes-Fourier equations fail to accurately predict important flow quantities since the amount of particle collisions is insufficient to maintain thermal equilibrium. By selecting different cutoff points in the Grad-moment-expansion of the Boltzmann equation, which can be understood as a Galerkin projection in velocity space, a nested model hierarchy consisting of moment equations is constructed. When linearized and augmented with suitable boundary conditions, these systems of equations are hyperbolic and energy-stable. Since the computational cost scales with the level of the model inside the hierarchy, it suggests itself to exploit anisotropy in the solution by locally adapting the model-level, such that an overall target accuracy is achieved more efficiently compared to the case of a single global model. In this work, we present an element-wise model-adaptive DG discretization. Neighboring elements with different model-levels are coupled by means of two half-Riemann problems embedded at each quadrature node of shared interfaces, enforcing a set of coupling conditions. The influence of different coupling conditions on the numerical solution and the energy stability of the discretization is investigated. The resulting scheme is embedded into a solve-estimate-mark-refine loop, where model-refined solutions are used to estimate the local model error. The efficacy of the present approach with respect to the needed number of degrees-of-freedom and runtime is evaluated by simulating a flow through a microchannel with curved obstacles. While the results show promise in reducing computational cost in settings where high model fidelity is required, further work on the coupling conditions and error estimation is required to increase the accuracy and robustness of the scheme.