The effect of horizontal photon transport within real-world clouds can be of consequence to remote sensing problems based on plane-parallel cloud models. In this paper, analytic approximations for the root-mean-square horizontal displacement of reflected and transmitted photons, relative to the incident cloud-top location, are derived for plane-parallel cloud layers. With anisotropic scattering, separate approximations are needed depending on the order of scattering. When sufficient numbers of photon scatterings occur, an approximation based on random walk theory (photon diffusion) is applicable; when scattering numbers are relatively small, a modification to the diffusion result is used. The resulting formulae are a function of the average number of photon scatterings, as well as particle asymmetry parameter and single scattering albedo. In turn, the average number of scatterings from plane-parallel, vertically inhomogeneous cloud layers can be determined from efficient adding/doubling radiative transfer procedures. The transport approximations are applied to liquid water clouds for typical remote sensing solar spectral bands, involving both conservative and non-conservative scattering. Results compare well with Monte Carlo calculations. Though the emphasis is on horizontal photon transport in terrestrial clouds, the derived approximations are applicable to general anisotropic, multiple scattering, plane-parallel radiative transfer problems. Approximations useful for three-dimensional transport are also given. The complete horizontal transport probability distribution can be described with an analytic distribution specified by the root-mean-square and average radial displacement values. However, it is shown empirically that the average displacement can be reasonably inferred from the root-mean-square value. An estimate for the horizontal transport distribution can then be made from the root-mean-square photon displacement alone.