We have run a total of 311 Direct Numerical Simulations (DNS) 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 wavenumber 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 DNS of forced unsteady turbulence and in wind tunnnel experiments of turbulent wakes and grid-generated turbulence to the classical scaling proposed by G.I. Taylor in 1935 and A.N. Kolmogorov in 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 thoughout the entire time-range of integration in all our DNS runs. The recently discovered dissipation scaling can be reformulated physically as a situation where the dissipation rates of the small and the large scales evolve together. We advance two hypotheses which may form the basis of a theoretical approach to unsteady turbulence cascades in the presence of large-scale coherent structures.