We perform a suite of high-resolution smoothed particle hydrodynamics simulations to investigate the orbital decay and mass evolution of massive black hole (MBH) pairs down to scales of ~30 pc during minor mergers of disk galaxies. Our simulation set includes star formation and accretion onto the MBHs, as well as feedback from both processes. We consider 1:10 merger events starting at z ~ 3, with MBH masses in the sensitivity window of the Laser Interferometer Space Antenna, and we follow the coupling between the merger dynamics and the evolution of the MBH mass ratio until the satellite galaxy is tidally disrupted. While the more massive MBH accretes in most cases as if the galaxy were in isolation, the satellite MBH may undergo distinct episodes of enhanced accretion, owing to strong tidal torques acting on its host galaxy and to orbital circularization inside the disk of the primary galaxy. As a consequence, the initial 1:10 mass ratio of the MBHs changes by the time the satellite is disrupted. Depending on the initial fraction of cold gas in the galactic disks and the geometry of the encounter, the mass ratios of the MBH pairs at the time of satellite disruption can stay unchanged or become as large as 1:2. Remarkably, the efficiency of MBH orbital decay correlates with the final mass ratio of the pair itself: MBH pairs that significantly increase their mass ratio are also expected to inspiral more promptly down to nuclear-scale separations. These findings indicate that the mass ratios of MBH pairs in galactic nuclei do not necessarily trace the mass ratios of their merging host galaxies but are determined by the complex interplay between gas accretion and merger dynamics.