In this work, we introduce a high-order domain decomposition method for the numerical solution of stiff Boltzmann-type equations. The method is designed to efficiently handle multiscale kinetic problems by dynamically identifying regions where the flow is close to thermodynamic equilibrium and regions where non-equilibrium effects are significant. Based on the dynamic detection of equilibrium and non-equilibrium regions, the proposed implementation automatically switches between the macroscopic model (e.g. Euler equations) and the kinetic model. The main challenge addressed in this work lies in the construction of a consistent and accurate coupling strategy between the macroscopic and kinetic solvers, particularly when high-order accuracy in time is desired. To address this issue, we develop a fully coupled implicit–explicit (IMEX) time integration strategy that operates across the decomposed subdomains and across the different models. The proposed IMEX coupling ensures stability in stiff regimes while preserving the global high-order temporal accuracy of the overall scheme. The resulting method combines the accuracy of kinetic solvers with the efficiency of macroscopic models, leading to a significant reduction in computational cost without sacrificing solution quality. The effectiveness and robustness of the approach are assessed through several numerical experiments in one and two spatial dimensions, which demonstrate correct model switching and the expected high-order convergence in time.