  The Particles on Demand method [B. Dorschner, F. B\"{o}sch and I. V. Karlin,
{\it Phys. Rev. Lett.} {\bf 121}, 130602 (2018)] was recently formulated with a
conservative finite volume discretization and validated against challenging
benchmarks. In this work, we rigorously analyze the properties of the reference
frame transformation and its implications on the accuracy of the model. Based
on these considerations, we propose strategies to boost the efficiency of the
scheme and to reduce the computational cost. Additionally, we generalize the
model such that it includes a tunable Prandlt number via quasi-equilibrium
relaxation. Finally, we adapt concepts from the multi-scale semi-Lagrangian
lattice Boltzmann formulation to the proposed framework, further improving the
potential and the operating range of the kinetic model. Numerical simulations
of high Mach compressible flows demonstrate excellent accuracy and stability of
the model over a wide range of conditions.
