We numerically investigate the characteristics of chaos evolution during wave packet spreading in two typical one-dimensional nonlinear disordered lattices: the Klein-Gordon system and the discrete nonlinear Schr{o}dinger equation model. Completing previous investigations cite{SGF13} we verify that chaotic dynamics is slowing down both for the so-called `weak and `strong chaos dynamical regimes encountered in these systems, without showing any signs of a crossover to regular dynamics. The value of the finite-time maximum Lyapunov exponent $Lambda$ decays in time $t$ as $Lambda propto t^{alpha_{Lambda}}$, with $alpha_{Lambda}$ being different from the $alpha_{Lambda}=-1$ value observed in cases of regular motion. In particular, $alpha_{Lambda}approx -0.25$ (weak chaos) and $alpha_{Lambda}approx -0.3$ (strong chaos) for both models, indicating the dynamical differences of the two regimes and the generality of the underlying chaotic mechanisms. The spatiotemporal evolution of the deviation vector associated with $Lambda$ reveals the meandering of chaotic seeds inside the wave packet, which is needed for obtaining the chaotization of the lattices excited part.
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