The paper presents the results of detection of sporadic changes (Forbush effects) in galactic cosmic ray variations during magnetic storms. The period of a weak magnetic storm occurred on 18 January 2017 (minimum Dst = −23 nT) and the period of a moderate magnetic storm occurred on 12 September 2023 (minimum Dst = −82 nT) are under analysis. The period preceding a strong magnetic storm occurred on 23 April 2023 (minimum Dst = −165 nT) is considered in detail. The theoretical basis of the investigation is the new method for detection of Forbush effects in cosmic ray variations based on neutron monitors data. Method bases, algorithm of its implementation are described and the method results are presented. Investigation results confirmed the efficiency of the suggested method. The detected sporadic changes had general character at all the stations, though the structures of these data are different. Both increases and decreases of cosmic ray intensities, exceeding the background level, were observed before the magnetic storms. The weak magnetic storm with gradual commencement was preceded by a long Forbush decrease. Anomalous increases of cosmic ray intensity were observed before the moderate and strong magnetic storms. Forbush effect intensities during the magnetic storms corresponded to the interplanetary disturbance intensities and the occurring magnetic storms strength. The investigation results confirm the importance of taking into account the galactic cosmic rays in the problems of space weather and applied significance of cosmic ray variation intensity measurements carried out by the ground network of neutron monitor stations.