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During an extensive multiwavelength campaign that we performed in 2013-2014, we found the prototypical Seyfert 1 galaxy NGC 5548 in an unusual condition of heavy and persistent obscuration. The newly discovered ``obscurer'' absorbs most of the soft X-ray continuum along our line of sight and lowers the ionizing luminosity received by the classical warm absorber. We present the analysis of the high resolution X-ray spectra collected with XMM-Newton and Chandra throughout the campaign, which are suitable to investigate the variability of both the obscurer and classical warm absorber. The time separation between these X-ray observations range from two days to eight months. On these timescales the obscurer is variable both in column density and in covering fraction. This is consistent with the picture of a patchy wind. The most significant variation occurred in September 2013 when the source brightened for two weeks. A higher and steeper intrinsic continuum and a lower obscurer covering fraction are both required to explain the spectral shape during the flare. We suggest that a geometrical change of the soft X-ray source behind the obscurer causes the observed drop in the covering fraction. Because of the higher soft X-ray continuum level, the September 2013 Chandra spectrum is the only X ray spectrum of the campaign in which individual features of the warm absorber could be detected. The spectrum shows absorption from Fe-UTA, OIV, and OV, consistent with belonging to the lower-ionization counterpart of the historical NGC 5548 warm absorber. Hence, we confirm that the warm absorber has responded to the drop in the ionizing luminosity caused by the obscurer.