We study high-dimensional conservation laws describing the transport of distribution functions, with the Vlasov equation (six dimensions plus time) as a key model in plasma physics. Solving such equations is computationally expensive due to their high dimensionality. Reduced-order models based on moments offer a less costly alternative while capturing essential physics. How- ever, these models require assumptions on the highest moment to close the system, known as the closure problem. Recently, Burby [1, equation 6] proposed an ansatz for the distribution function which parametrizes it in terms of a finite collection of moments and developed closures with Hamiltonian structure. These closures are exact solutions of the Vlasov-Poisson model when the distribution function aligns with the ansatz. Using the ansatz due to Burby, we present exact reduced models of degree zero, one, and two for a class of high-dimensional conservation laws. We demonstrate that these models are exact solutions of the conservation laws in a distributional sense. By enriching Burby’s ansatz with the particle-in-cell ansatz, we propose hybrid particle-fluid models that preserve the same exactness property. For the degree zero model, we demonstrate dynamic adaptivity between fluid and particle repre- sentations. In plasma applications, the models extend Burby’s degree zero, one, and two models from the Vlasov-Poisson system to the Vlasov-Maxwell system. The degree zero hybrid model aligns with Tronci’s kinetic Hamiltonian hybrid model for the Vlasov-Maxwell system when the kinetic component is discretized with particles [2]. Additionally, we apply these models to relativistic Vlasov-Maxwell and collisional plasmas. For the degree zero relativistic model, we develop energy- and mass-conserving discretizations using its Hamiltonian structure.