We extend previous work using photon diffusion theory to describe radiative transfer through dense plane-parallel clouds at non-absorbing wavelengths. The focus here is on the scaling of space- and time-domain moments for transmitted light with respect to cloud thickness H and optical depth tau; the new results are: accurate prefactors for asymptotic scaling, pre-asymptotic correction terms in closed form, 3D effects for internal variability in tau, and the root-mean-square (RMS) transit time or pathlength. Mean path length is proportional to H for dimensional reasons and, from random-walk theory, we already know that it is also proportional to (1–g)tau for large enough tau (g being the asymmetry factor). Here we show that the prefactor is precisely 1/2 and that corrections are significant for (1–g)tau < 10, which includes most actual boundary-layer clouds. We also show that RMS path length is not much larger than the mean for transmittance (its prefactor is sqrt(7/20) ~ 0.59); this proves that, in sharp contrast with reflection, path length distributions are quite narrow in transmission. If the light originates from a steady point-source on a cloud boundary then a fuzzy spot is observed on the opposite boundary. This problem is formally mapped to the pulsed-source problem and we show that the RMS radius of this spot slowly approaches sqrt(2/3)H as tau increases; we also show that the transmitted spot shape has a flat top and an exponential tail. Because all pre-asymptotic corrections are computed here, our diffusion results are accurate when compared to Monte Carlo counterparts for tau > 5 whereas the classic scaling relations apply only for tau > 70, assuming g = 0.85. The temporal quantities shed light on observed absorption properties and optical lightning waveforms. The spatial quantity controls the three-dimensional radiative smoothing process in transmission which was recently observed in spectral analyses of time-series of zenith radiance at 725 nm. We describe opportunities in ground-based cloud remote sensing using the new developments and illustrate with simulations of 3D solar radiative transfer in realistic models of stratocumulus. Finally, since this analytical diffusion study applies only to weakly variable stratus layers, we discuss extensions to more complex cloud systems using anomalous diffusion theory