Crossed beam energy transfer, CBET, in high-intensity laser–plasma interaction is investigated for the case of optically smoothed laser beams. In the two approaches to laser-driven inertial confinement fusion experiments, the direct-drive and the indirect-drive, CBET is of great importance because it governs the coupling of laser energy to the plasma. We use the two-dimensional wave-coupling code H armony to simulate the transfer between two laser beams with speckle structure that overlap in a plasma with an inhomogeneous flow profile. We compare the CBET dynamics for laser beams with spatial incoherence and with spatio-temporal incoherence; in particular we apply the smoothing techniques using random phase plates (RPPs) and smoothing by spectral dispersion (SSD), respectively. It is found that for laser beams (wavelength λ 0 ) with intensities ( I L ) above I L  ∼ 2 × 10 15  W cm −2 ( λ 0 /0.35 µm) −2 ( T e /keV), both the so-called plasma-induced smoothing as well as self-focusing in intense laser speckles induce temporal incoherence; the latter affects the CBET and the angular distribution of the light transmitted behind the zone of beam overlap. For RPP-smoothed incident beams, the resulting band width of the transmitted light can already be of the same order as the effective band width of the SSD available at major laser facilities. We examine the conditions when spatio-temporal smoothing techniques become efficient for CBET. This article is part of a discussion meeting issue ‘Prospects for high gain inertial fusion energy (part 1)’.