[Abridged] 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 FRII objects (kinetic luminosities, lifetimes, and the central densities of their environments). In particular, the radio and kinetic luminosity functions of FRIIs are investigated using the complete, flux limited radio catalogues of 3CRR and Best et al. We construct multidimensional Monte Carlo simulations using semi-analytical models of FRII radio source growth 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 fundamental source parameters. We allow the source physical properties 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 FRIIs with those determined from radiative luminosities of AGN 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 FRIIs at low redshifts, or a transition to a kinetically-dominated, radiatively-inefficient FRII population.
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