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It has been proposed that mergers induce starbursts and lead to important morphological changes in galaxies. Most studies so far have focused on large galaxies, but dwarfs might also experience such events, since the halo mass function is scale-free in the concordance cosmological model. Notably, because of their low mass, most of their interactions will be with dark satellites. In this paper we follow the evolution of gas-rich disky dwarf galaxies as they experience a minor merger with a dark satellite. We aim to characterize the effects of such an interaction on the dwarf's star formation, morphology, and kinematical properties. We performed a suite of carefully set-up hydrodynamical simulations of dwarf galaxies that include dark matter, gas, and stars merging with a satellite consisting solely of dark matter. For the host system we vary the gas fraction, disk size and thickness, halo mass, and concentration, while we explore different masses, concentrations, and orbits for the satellite. We find that the interactions cause strong starbursts of both short and long duration in the dwarfs. Their star formation rates increase by factors of a few to 10 or more. They are strongest for systems with extended gas disks and high gas fractions merging with a high-concentration satellite on a planar, radial orbit. In contrast to analogous simulations of Milky Way-mass galaxies, many of the systems experience strong morphological changes and become spheroidal even in the presence of significant amounts of gas. The simulated systems compare remarkably well with the observational properties of a large selection of irregular dwarf galaxies and blue compact dwarfs. This implies that mergers with dark satellites might well be happening but not be fully evident, and may thus play a role in the diversity of the dwarf galaxy population., Comment: 15 pages (+4 as appendix), 17 figures (+5 in appendix). A&A accepted