We have run a total of 311 direct numerical simulations (DNSs) of decaying three-dimensional Navier-Stokes turbulence in a periodic box with values of the Taylor length-based Reynolds number up to about 300 and an energy spectrum with a wide wave-number range of close to -5/3 power-law dependence at the higher Reynolds numbers. On the basis of these runs, we have found a critical time when (i) the rate of change of the square of the integral length scale turns from increasing to decreasing, (ii) the ratio of interscale energy flux to high-pass filtered turbulence dissipation changes from decreasing to very slowly increasing in the inertial range, (iii) the signature of large-scale coherent structures disappears in the energy spectrum, and (iv) the scaling of the turbulence dissipation changes from the one recently discovered in DNSs of forced unsteady turbulence and in wind tunnel experiments of turbulent wakes and grid-generated turbulence to the classical scaling proposed by G. I. Taylor [Proc. R. Soc. London, Ser. A 151, 421 (1935)1364-502110.1098/rspa.1935.0158] and A. N. Kolmogorov [Dokl. Akad. Nauk SSSR 31, 538 (1941)]. Even though the customary theoretical basis for this Taylor-Kolmogorov scaling is a statistically stationary cascade where large-scale energy flux balances dissipation, this is not the case throughout the entire time range of integration in all our DNS runs. The recently discovered dissipation scaling can be reformulated physically as a situation in which the dissipation rates of the small and large scales evolve together. We advance two hypotheses that may form the basis of a theoretical approach to unsteady turbulence cascades in the presence of large-scale coherent structures.