The interaction of electrons with a flux of ballistic phonons leads to excitation of many angular harmonics of an electron distribution function. We show that this property dramatically modifies the magnetothermoelectric phenomena in two-dimensional electron systems with boundaries. By considering classical magnetotransport of electrons in a narrow channel with partly diffusive boundary scattering, we show that the phonon flux excites a pattern of current stripes with alternating directions of propagation along the channel. The Nernst voltage due to phonon drag appears already in the classical transport regime and can be comparable with the Seebeck voltage, while the latter acquires a dependence on the magnetic field. The temperature dependence of these voltages shows an unusual behavior determined by relaxation of higher-order harmonics of the distribution function via electron-electron scattering. Our experimental studies of mesoscopic samples based on high-quality GaAs quantum wells confirm the main properties of the thermoelectric response suggested by the theory.