In the statistical quasilinear theory of weak plasma turbulence, charged particles moving in electrostatic fluctuations diffuse in velocity, i.e. the velocity variance 〈Δv2(t)〉 increases linearly with time t, for times long compared with the auto-correlation time τac of the field, which may be estimated as the reciprocal of the spectral width of the fluctuations. Recent test-particle simulations have revealed a new regime at very long timescales t[Gt ]τac where quasilinear theory breaks down, for intermediate field amplitudes. As this behaviour is not consistent with a diffusion on quasilinear timescales, the problem of the motion of particles in a broadband wave field, for the case of a slowly growing field, is considered here from a purely dynamical point of view, introducing no statistics on the field and no restriction on the amplitude of this field. By determining, on a given timescale, and in the frame of wave–particle interaction, the spectral width over which waves interact efficiently with a particle, a new timescale is found: the nonlinear time of wave–particle interaction τNL∝ (spectral density of energy)−1/3[Gt ]τac. This is the correlation time of the dynamics. For times shorter than τNL, the particles trajectories remain globally regular, and do not separate: they follow a quasifractal set of dimension 2. For times long compared with τNL, there appears a ‘true’ diffusive regime with mixing and decorrelation, due to nonlinear mixing in phase space and the localization of the wave–particle interaction. These theoretical results are confirmed by a numerical study of the velocity variance as a function of time. In particular, the particle dynamics really do become diffusive on timescales several orders of magnitude longer than that predicted by quasilinear theory (namely [Gt ]τNL[Gt ]τac). Finally, deviations from the quasilinear value of the diffusion coefficient and wave growth rate, discussed in the literature, are explained.