The kinematic properties of the gaseous and stellar components of 11 ultraluminous infrared galaxies (ULIRGs; 14 nuclei) are investigated by means of integral field spectroscopy (IFS) with the INTEGRAL system and available IR and CO millimeter spectroscopy. The sample of ULIRGs cover different phases of the merging process and span all levels of activity from pure starbursts to Seyfert nuclei. The IFS data show that the ionized gas has a complex velocity structure with peak-to-peak velocity differences of a few to several hundred km s-1, detected in tidal tails or extranuclear star-forming regions. The velocity field of the ionized gas on scales of a few to several kiloparsecs is dominated by tidally induced flows and does not, in general, correspond to rotationally supported systems with a privileged orientation along the major rotating axis. The central velocity amplitude of the ionized gas and stars shows discrepancies in some galaxies but has, on average, a similar value (ratio of 0.92+/-0.37) , while the velocity amplitude of the molecular gas is a factor of 2 larger (ratio of 1.9+/-0.6) than that of the stars and ionized gas. The central velocity amplitude measured using different kinematic tracers should therefore not be used in ULIRGs as a reliable tracer of mass, in general. The IFS data also show that the velocity dispersion of the ionized gas maps the large-scale motions associated with tidal tails and extranuclear regions, with often the highest velocity dispersion not being associated with the nucleus galaxies. There is, however, a good agreement between the central ionized gas and stellar velocity dispersions (ratio of 1.01+/-0.13), while the cold molecular gas velocity dispersion has lower values (average of about 0.8 that of the stellar and ionized gas). The central ionized gas velocity dispersion is therefore a robust and homogeneous observable and a good tracer of the dynamical mass in these systems. The IFS-based central ionized gas velocity dispersion measurements confirm that ULIRGs' hosts are moderate-mass (<=m*) galaxies, as previously concluded by Tacconi and coworkers. In general, velocity amplitudes should not be used to estimate the dynamical mass in high-z star-forming systems, such as Lyman break and in particular submillimeter galaxies, since they show irregular stellar and gaseous structures similar to those present in low-z merging systems such as ULIRGs, the subject of this study. A more reliable method is to measure the central velocity dispersion using the strong, high equivalent width, rest-frame optical emission lines, provided the location of the nucleus is independently established by high angular resolution red or near-IR rest-frame imaging. The kinematics derived from the millimeter CO line suggest that the cold gas in ULIRGs does not share the velocity field of the stars and ionized gas and seems to be more rotationally supported. This result needs to be investigated in more detail with a larger sample of low-z ULIRGs before using the millimeter CO line widths as a dynamical mass tracer in high-z submillimeter galaxies. Based on observations with the William Herschel Telescope operated on the island of La Palma by the ING in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Based also on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.