Radio galaxies and quasars are among the largest and most powerful single objects known and are believed to have had a significant impact on the evolving Universe and its large-scale structure. We explore the intrinsic and extrinsic properties of the population of Fanaroff–Riley type II (FR II) objects, i.e. their kinetic luminosities, lifetimes and the central densities of their environments. In particular, the radio and kinetic luminosity functions of these powerful radio sources are investigated using the complete, flux-limited radio catalogues of the Third Cambridge Revised Revised Catalogue (3CRR) and Best et al. We construct multidimensional Monte Carlo simulations using semi-analytical models of FR II source time evolution to create artificial samples of radio galaxies. Unlike previous studies, we compare radio luminosity functions found with both the observed and simulated data to explore the best-fitting fundamental source parameters. The new Monte Carlo method we present here allows us to (i) set better limits on the predicted fundamental parameters of which confidence intervals estimated over broad ranges are presented and (ii) generate the most plausible underlying parent populations of these radio sources. Moreover, as has not been done before, we allow the source physical properties (kinetic luminosities, lifetimes and central densities) to co-evolve with redshift, and we find that all the investigated parameters most likely undergo cosmological evolution. Strikingly, we find that the break in the kinetic luminosity function must undergo redshift evolution of at least (1 + z)3. The fundamental parameters are strongly degenerate, and independent constraints are necessary to draw more precise conclusions. We use the estimated kinetic luminosity functions to set constraints on the duty cycles of these powerful radio sources. A comparison of the duty cycles of powerful FR IIs with those determined from radiative luminosities of active galactic nuclei of comparable black hole mass suggests a transition in behaviour from high to low redshifts, corresponding to either a drop in the typical black hole mass of powerful FR IIs at low redshifts, or a transition to a kinetically dominated, radiatively inefficient FR II population.