Bessel-Gaussian beams have potential applications in laser atmospheric engineering applications such as directed energy and space optical communication. The influence of the thermal blooming effect on composite Bessel-Gaussian (cBG) beams propagating in the atmosphere is studied using the multi-phase screen method and the fast Fourier transform to solve the thermal blooming equation. The Fast Fourier Transform (FFT) is capable of processing the propagation between the phase screens. Consequently, the propagation process of the cBG beam from z_n to z_n+1 can be divided into three steps: the first step is the vacuum propagation Δz/2 via the fast Fourier transform; The second step is the phase change caused by the atmospheric refractive index fluctuations; The last step is the remaining vacuum propagation Δz/2 via the fast Fourier transform again. Thus, the atmospheric propagation of the beam is converted into the propagation between the multi-phase screens, and the phase screens represent the disturbance of thermal blooming on the beam. Based on the above method, a 4D computer code is designed to simulate the time-dependent propagation of cBG beams in the atmosphere. To be convenient to analyze the thermal blooming effect of cBG beams, we provide the expression for the amplitude factor. Considering both the diffraction of the optical field and the time scale of the thermal blooming effect, we study the beam distortion, orbital angular momentum spectrum, and mode crosstalk of cBG beams under the thermal blooming effect. Due to the wind-dominated thermal blooming effect, the phase singularity positions cBG beams are irregularly shifted. By analyzing the orbital angular momentum spectrum of the cBG beam under the thermal blooming effect, we found that the energy between the initial modes of the beam is transferred to each other, producing mode crosstalk.With the increase of propagation distance or the decrease of crossing wind velocity, the strength of the thermal blooming effect increases, resulting in the enhancement of mode crosstalk. The relative crosstalk energy of the cBG beams gradually increases with time; until the absorption of beam power by the atmosphere is balanced by the heat exchange caused by the crossing wind, the thermal blooming effect reaches a steady state. The relative crosstalk energy no longer changes with time. The flow rate of a medium determines the speed of its energy transport. The thermal effects during beam propagation are stronger for lower wind speeds, resulting in stronger mode crosstalk. The relative crosstalk energy decreases with the difference value of an initial angular quantum number. Therefore, the cBG beam with a larger difference value of an initial angular quantum number has the weaker mode crosstalk affected by thermal blooming. The thermal blooming effect of rotating cBG beams is studied. A rotating cBG beam can spread uniformly in the atmosphere due to the thermal blooming effect controlled by the wind. Moreover, the mode crosstalk of rotating cBG beams is smaller than that of non-rotating cBG beams. The mode crosstalk of rotating cBG beams decreases with the difference value of radial wave number. To sum up, increasing the differences values of the initial angular quantum number and radial wave number can effectively reduce the mode crosstalk of cBG beams. The results obtained in this paper have important implications for the applications of BG beams in laser atmospheric engineering, such as directed energy and space optical communication.