In this article, the results of investigation of processes occurring during the molecular-beam deposition of germanium layers on Si$_3$N$_4$ dielectric substrates within a wide range of the Ge film growth temperatures (30 to 600{\deg}C) are presented. The intensity of the IR absorption bands related to the vibrations of the N$-$H and Si$-$N bonds are established to decrease with the increase of the Ge deposition temperature. It appears that this phenomenon cannot be explained only as a thermally activated process. Simultaneously, the peak corresponding to the Ge$-$N vibration bonds emerges in the X-ray photoelectron spectra. We suppose that the deposition of germanium layers on Si3N4 dielectric substrates containing hydrogen atoms causes the diffusion of hydrogen atoms from the dielectric layer into the growing film. The experimental results may be interpreted in terms of a model, according to which the migration of hydrogen atoms from the Si$_3$N$_4$ layer into the growing germanium film is due to the difference in chemical potentials of hydrogen atoms in the dielectric layer and the germanium film. This process initiates the diffusion of germanium atoms in the opposite direction, into the Si$_3$N$_4$ layer, where they connect to free bonds of nitrogen atoms arising due to the escape of hydrogen atoms. The analogous processes occur during the deposition of silicon layers on Si$_3$N$_4$ substrates.