Context: The origin of supergranulation has not been understood yet. Contradictory results have been obtained in the past concerning the relation between supergranule properties (mostly cell size) and the solar cycle. Aims: We propose to study the variation in supergranule cell sizes and velocity fields over the solar cycle, as well as the intensity variation inside supergranules. Methods: We define supergranule cells from maps of horizontal velocity field divergences. The flow fields are derived from MDI/SOHO intensity maps. The intensity variation in supergranules are compared to the variation inside granules resulting from a numerical simulation. The variation in these profiles with the cell size and over the solar cycle is also analysed. Results: We find that cell sizes are smaller on average at cycle maximum. We also find that the slope between Log (V_rms) and Log (R) is weakly correlated with the spot number (i.e. the global activity level) but anti-correlated with the local magnetic field. We also confirm the decrease in the intensity variation from cell centre to the boundary, which puts a lower limit on the temperature variation of 0.57 ± 0.06 K. This temperature difference is of 1.03 ± 0.05 K when considering the areas of strongest divergence and strongest convergence. We observe a strong similarity in the intensity variation inside supergranules and granules. A significant variation with the cell size is observed, also similar to that in granules, but the variation over the solar cycle is not significant. Conclusions: The sign of the variation in supergranule cell sizes over the solar cycle is in agreement with what can be expected from the influence of the magnetic field. The observed intensity variations show that a common process could be the origin of both granules and supergranules.