Data consistency conditions (DCCs) express the redundancy in the projections. In X-ray computed tomography, the most common conditions are expressed pairwise on the projections or as equality between projection-based moments and polynomials. The latter is better known in the parallel-beam geometry as the Helgason-Ludwig consistency conditions (HLCCs). The DCCs are often used to self-calibrate radiography systems. In this paper, we adjust data consistency conditions to a time-dependent model of the data in the parallel linogram geometry. We show that it is not possible to estimate the parameters of a uniform motion of a translating object using the DCCs. However, we show that we can estimate the average speed with prior information on the object's center of mass. Then, we model and estimate the parameters of a periodical variation of the motion. Finally, we run simulations to assess the performances of our method.