(Abridged) Processes driving mass assembly are expected to evolve on different timescales along cosmic time. A transition might happen around z ~ 1 as the cosmic star formation rate starts its decrease. Identifying the dynamical nature of galaxies on a representative sample is necessary to infer and compare the mass assembly mechanisms across cosmic time. We present an analysis of the kinematics properties of 50 galaxies with 0.9 < z < 1.6 from the MASSIV sample observed with SINFONI/VLT with 4.5x10^9 Msun < M < 1.7x10^11 Msun and 6 Msun/yr < SFR < 300 Msun/yr. This is the largest sample with 2D-kinematics in this redshift range. We provide a classification based on kinematics as well as on close galaxy environment. We find that 29% of galaxies are experiencing merging or have close companions that may be gravitationally linked. This is placing a lower limit on the fraction of interacting galaxies. We find that at least 44% of the galaxies display ordered rotation whereas at least 35% are non-rotating objects. All rotators except one are compatible with rotation-dominated (Vmax/sigma > 1) systems. Non-rotating objects are mainly small objects (Re < 4 kpc). Combining our sample with other 3D-spectroscopy samples, we find that the local velocity dispersion of the ionized gas component decreases continuously from z ~ 3 to z = 0. The proportion of disks also seems to be increasing in star-forming galaxies when the redshift decreases. The number of interacting galaxies seems to be at a maximum at z ~ 1.2. These results draw a picture in which cold gas accretion may still be efficient at z ~ 1.2 but in which mergers may play a much more significant role at z ~ 1.2 than at higher redshift. From a dynamical point of view, the redshift range 1 < z < 2 therefore appears as a transition period in the galaxy mass assembly process.
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