We use the Sloan Digital Sky Survey Data Release 12, which is the largest available white dwarf catalogue to date, to study the evolution of the kinematical properties of the population of white dwarfs in the Galactic disc. We derive masses, ages, photometric distances and radial velocities for all white dwarfs with hydrogen-rich atmospheres. For those stars for which proper motions from the USNO-B1 catalogue are available, the true three-dimensional components of the stellar space velocity are obtained. This subset of the original sample comprises 20 247 objects, making it the largest sample of white dwarfs with measured three-dimensional velocities. Furthermore, the volume probed by our sample is large, allowing us to obtain relevant kinematical information. In particular, our sample extends from a Galactocentric radial distance R_G_=7.8 to 9.3kpc, and vertical distances from the Galactic plane ranging from Z=-0.5 to 0.5kpc. We examine the mean components of the stellar three-dimensional velocities, as well as their dispersions with respect to the Galactocentric and vertical distances. We confirm the existence of a mean Galactocentric radial velocity gradient, {partial}<V_R_>/{partial}R_G_=-3+/-5km/s/. We also confirm north-south differences in <V_z_> . Specifically, we find that white dwarfs with Z>0 (in the North Galactic hemisphere) have <V_z_><0, while the reverse is true for white dwarfs with Z<0. The age-velocity dispersion relation derived from the present sample indicates that the Galactic population of white dwarfs may have experienced an additional source of heating, which adds to the secular evolution of the Galactic disc.