Transmissions through low-dimensional disordered mesoscopic systems, subject to a perpendicular magnetic field and random spin-orbit coupling, are studied. The transmissions are obtained from the scattering matrix, in whose derivation the role of spin-orbit coupling as a phase factor is emphasized. The scattering matrix of a system connected to several reservoirs by single-channel leads is considered; for the one-dimensional ring, its form is derived analytically. In the latter case, it is found that the point-to-point transmission is an even function of the flux only when there is no additional connection of the system to an external reservoir. Otherwise, the transmission is a general (periodic) function of the flux. Implications of this observation are discussed. Averaging over an ensemble of realizations, it is shown that the transmission of a two-dimensional sample includes only even harmonics as a function of the magentic flux per unit cell when the average over the spatial disorder alone is insufficient to extinguish the odd harmonics. The transmission through a one-dimensional ring is shown analytically to consist solely of the zeroth and the second harmonics of the threading flux.