We present here a study of propagation of a plane vertically directed laser beam through a gravitationally convective flow of an absorbing medium. The optical beam is considered in the paraxial approximation of the wave theory. The gravitational convection of the medium is treated based on Boussinesq equation. The comparison between regimes of the strong (developed) convection and moderate (viscous) one is made. The profiles of intensity, velocity and temperature are demonstrated for bottom, middle and top sections of the absorbing medium volume under study. The effect of diffraction and thermal blooming on the laser beam propagating under transient moderate (viscous) free convection regime is investigated.