In this paper, a new method for the mathematical tomographic reconstruction of time-varying objects (4D tomography) is presented. The method is applicable under the conditions that the amount of substance at each spatial point does not decrease, and only one projection image is available per time point during the process of the object changing. The temporal resolution of the proposed method is obtained by using an iterative approach to reconstruction. Prior knowledge about object morphology is exploited to improve the convergence of the algorithm by reducing the number of processed spatial points. Verification of the algorithm operation was carried out using model examples. The presented method was experimentally tested on the tomographic reconstruction of the dynamics of the capillary rise of a liquid. Our experiments demonstrated that monitoring fast processes, when the speed of the process does not allow to measure more than one tomographic projection, is potentially possible.