Context. Variability across the electromagnetic spectrum is a property of active galactic nuclei (AGN) that can help constrain the physical properties of these galaxies. Nonetheless, the way in which the changes happen and whether they occur in the same way in every AGN are still open questions. Aims. This is the third in a series of papers with the aim of studying the X-ray variability of different families of AGN. The main purpose of this work is to investigate the variability pattern(s) in a sample of optically selected Seyfert 2 galaxies. Methods. We use the 26 Seyfert 2s in the Véron-Cetty and Véron catalog with data available from Chandra and/or XMM-Newton public archives at different epochs, with timescales ranging from a few hours to years. All the spectra of the same source were simultaneously fitted, and we let different parameters vary in the model. Whenever possible, short-term variations from the analysis of the light curves and/or long-term UV flux variations were studied. We divided the sample into Compton-thick and Compton-thin candidates to account for the degree of obscuration. When transitions between Compton-thick and thin were obtained for different observations of the same source, we classified it as a changing-look candidate. Results. Short-term variability at X-rays was studied in ten cases, but variations are not found. From the 25 analyzed sources, 11 show long-term variations. Eight (out of 11) are Compton-thin, one (out of 12) is Compton-thick, and the two changing-look candidates are also variable. The main driver for the X-ray changes is related to the nuclear power (nine cases), while variations at soft energies or related to absorbers at hard X-rays are less common, and in many cases these variations are accompanied by variations in the nuclear continuum. At UV frequencies, only NGC 5194 (out of six sources) is variable, but the changes are not related to the nucleus. We report two changing-look candidates, MARK 273 and NGC 7319. Conclusions. A constant reflection component located far away from the nucleus plus a variable nuclear continuum are able to explain most of our results. Within this scenario, the Compton-thick candidates are dominated by reflection, which suppresses their continuum, making them seem fainter, and they do not show variations (except MARK 3), while the Compton-thin and changing-look candidates do.