Recently, a new family of autoionization processes has been identified in aqueous phases. The processes are initiated by core-electron ionization of a solute molecule and involve proton transfer along the solute-solvent hydrogen bond. As a result, short-lived singly charged cations form with structures sharing a proton between solute and solvent molecules. These molecular transients decay by autoionization, which creates reactive dicationic species with the positive charges delocalized over the entire molecular entity. Here, we investigate the ultrafast electron and nuclear dynamics following the core ionization of hydrated ammonia and glycine. Both molecules serve as models for exploring the possible role of the nonlocal relaxation processes in the chemical reactivity at the interface between, for instance, a protein surface and aqueous solution. The nature of the postionization dynamical processes is revealed by high-accuracy Auger-electron spectroscopy measurements on liquid microjets in vacuum. The proton-transfer-mediated processes are identified by electron signals in the high-energy tail of the Auger spectra with no analogue in the Auger spectra of the corresponding gas-phase molecule. This high-energy tail is suppressed for deuterated molecules. Such an isotope effect is found to be smaller for aqueous ammonia as compared to the hydrated H2O molecule, wherein hydrogen bonds are strong. An even weaker hydrogen bonding for the hydrated amino groups in glycine results in a negligibly small proton transfer. The dynamical processes and species formed upon the nitrogen-1s core-level ionization are interpreted using methods of quantum chemistry and molecular dynamics. With the assistance of such calculations, we discuss the conditions for the proton-transfer-mediated relaxation processes to occur. We also consider the solvent librational dynamics as an alternative intermolecular ultrafast relaxation pathway. In addition, we provide experimental evidence for the umbrella-type motion in aqueous ammonia upon core ionization. This intramolecular channel proceeds in parallel with intermolecular relaxation processes in the solution.