The forthcoming Laser Interferometer Space Antenna (LISA) will probe the population of coalescing massive black hole (MBH) binaries up to the onset of structure formation. Here, we simulate the galactic-scale pairing of ∼106 M⊙ MBHs in a typical, non-clumpy main-sequence galaxy embedded in a cosmological environment at z = 7–6. In order to increase our statistical sample, we adopt a strategy that allows us to follow the evolution of six secondary MBHs concomitantly. We find that the magnitude of the dynamical-friction-induced torques is significantly smaller than that of the large-scale, stochastic gravitational torques arising from the perturbed and morphologically evolving galactic disc, suggesting that the standard dynamical friction treatment is inadequate for realistic galaxies at high redshift. The dynamical evolution of MBHs is very stochastic, and a variation in the initial orbital phase can lead to a drastically different time-scale for the inspiral. Most remarkably, the development of a galactic bar in the host system either significantly accelerates the inspiral by dragging a secondary MBH into the centre, or ultimately hinders the orbital decay by scattering the MBH in the galaxy outskirts. The latter occurs more rarely, suggesting that galactic bars overall promote MBH inspiral and binary coalescence. The orbital decay time can be an order of magnitude shorter than what would be predicted relying on dynamical friction alone. The stochasticity and the important role of global torques have crucial implications for the rates of MBH coalescences in the early Universe: both have to be accounted for when making predictions for the upcoming LISA observatory.