Discrete particles released into turbulent carrier flows respond to that turbulence over periods of time which depend on the particle inertia and the initial state of the particle. Due to the turbulence, the velocity of the particles becomes random and they will become dispersed throughout the carrier flow. The level of randomness of the particle velocities can be quantified by the particle kinetic stress and the spreading rate is characterized by the dispersion coefficient. In the idealized case of stationary isotropic turbulence, the kinetic stress and dispersion coefficients approach limiting values long after release. Previous analysis by the author has provided development times for dispersion coefficients and kinetic stress in cases where particles were released from a point source: (i) from an initial state of rest, and (ii) with kinetic stress identical to that found in the steady state. This paper generalizes the analysis by allowing for arbitrary initial particle velocity and displacement distributions. As with the previous analysis, the primary focus is on high-inertia particles. Unlike the previous work, however, the present analysis is applicable to the common practice in direct numerical simulations of setting initial particle velocities equal to local fluid velocities. Several different estimators of the particle dispersion coefficient are considered. Whereas each estimator leads to the same long-time value, the time taken to approach this value can vary dramatically. Expressions for development times are given for the particle kinetic stress and dispersion coefficients. Consequences of the analysis for experimental and computational studies are discussed.