The reorganization of the environment of a self-trapped proton in a polarizable medium is a necessary step making proton transfer possible, in addition to the decreasing distance between the donor and the acceptor. We focus here on the mechanisms of proton transfer (PT) in the cubic perovskite oxide BaZrO3. From density functional theory (DFT) calculations, we show that the self-trapping distortion, as well as the reorganization motion in this compound mostly originate from a rotation/deformation of the four oxygen octahedra surrounding the proton. This identification allows to extend the Landau-Zener curve-crossing approach describing PT in solutions with polar solvents [D.Borgis, J.T. Hynes, J.Phys. Chem. 100, 1118 (1996)] to this system. Following the idea that PT is quantum by nature, the process is described as resulting from thermal fluctuations of (classical) lattice coordinates, among which oxygen octahedral rotations/deformations play the same role as the solvent reorganization in a solution: this reorganization allows the system to reach coincidence configurations, in which the protonic ground levels in the initial and final wells are equalized, resulting in a possible transfer, with a probability given by the Landau-Zener formula. The PT rate is expressed as a sum of contributions over all possible coincidence configurations, treating with the same formalism ground-state non adiabatic tunneling transfers, adiabatic transfers in which the proton zero-point energy lies above the energy barrier in the proton coordinate, and all other intermediate situations. A simplified model is proposed, in which the energy of the system depends on two relevant lattice distortions, (i) the amplitude S of the reorganization motion (which appears as the natural reaction coordinate for PT), and (ii) the distance Q between the two oxygens involved in the transfer. DFT calculations are used to feed the model, and calculate the contribution of each coincidence configuration to the transfer rate. Within this model, the slow dynamics of the reorganization (typical vibration frequency 3-5 THz), combined with the soft profile of the coincidence energy as a function of Q, make adiabatic transfers (and tunneling with large transfer probability) dominate down to low temperature (via almost "barrierless" coincidence configurations with short oxygen-oxygen separation), excluding the possibility of a non-adiabatic tunneling regime.