The establishment of the hydrodynamic regime is analysed, introducing equations with time-dependent as well as space-dependent transport coefficients and deriving the relations between the two sets of coefficients. Based upon the Fourier transformation technique, the short-time behaviour of swarms in idealized free space is investigated, starting from the Boltzmann equation by means of analysis of its eigenvalue problem. The relaxation times of time-dependent transport coefficients for the establishment of the hydrodynamic regime and the corresponding relaxation distances of space-dependent coefficients, which are defined from the position dependence of the arrival-time spectra and its time moments, have been formulated. By using the theoretical results, which were rewritten in matrix form using the Burnett basis function, the relaxation times of ionization growth, drift and diffusion have been calculated for electrons in krypton gas as functions of the reduced electric field and the mean energy of the initial Gaussian velocity distribution. The relaxation distances d0 in the measurement of the Townsend first ionization coefficient in krypton have been calculated.