Adaptive moving mesh research usually focuses either on analytical derivations for prescribed solutions or on pragmatic solvers with challenging physical applications. In the latter case, the monitor functions that steer mesh adaptation are often defined in an ad-hoc way. In this paper we generalize our previously used monitor function to a balanced sum of any number of monitor components. This avoids the trial-and-error parameter fine-tuning that is often used in monitor functions. The key reason for the new balancing method is that the ratio between the maximum and average value of a monitor component should ideally be equal for all components. Vorticity as a monitor component is a good motivating example for this. Entropy also turns out to be a very informative monitor component. We incorporate the monitor function in an adaptive moving mesh higher-order finite volume solver with HLLC fluxes, which is suitable for nonlinear hyperbolic systems of conservation laws. When applied to compressible gas flow it produces very sharp results for shocks and other discontinuities. Moreover, it captures small instabilities (Richtmyer-Meshkov, Kelvin-Helmholtz). Thus showing the rich nature of the example problems and the effectiveness of the new monitor balancing.