We provide a sharp generalization to the nonautonomous case of the well-known Kobayashi estimate for proximaliterates associated with maximal monotone operators. We then derive a bound for the distance between acontinuous-in-time trajectory, namely the solution to the differential inclusion $\dot{x} + A(t)x $∋ $ 0$, and thecorresponding proximal iterations. We also establish continuity properties with respect to time of the nonautonomousflow under simple assumptions by revealing their link with the function $t \mapsto A(t)$. Moreover, our sharperestimations allow us to derive equivalence results which are useful to compare the asymptotic behavior of thetrajectories defined by different evolution systems. We do so by extending a classical result of Passty to thenonautonomous setting.