We searched for thermonuclear X-ray bursts from Galactic neutron stars in all event mode data of the Neil Gehrels Swift Observatory collected until March 31, 2018. In particular, we are interested in the intermediate-duration bursts (shell flashes fueled by thick helium piles) with the ill-understood phenomenon of strong flux fluctuations. Nine such bursts have been discussed in the literature to date. Swift is particularly suitable for finding additional examples. We find and list a total of 134 X-ray bursts; 44 are detected with BAT only, 41 with XRT only, and 49 with both. Twenty-eight bursts involve automatic slews. We find 12 intermediate-duration bursts, all detected in observations involving automatic slews. Five show remarkably long Eddington-limited phases in excess of 200 s. Five show fluctuations during the decay phase; four of which are first discussed in the present study. We discuss the general properties of the fluctuations, considering also 7 additional literature cases. In general two types of fluctuations are observed: fast ones, with a typical timescale of 1 s and up and downward fluctuations of up to 70%, and slow ones, with a typical timescale of 1 min and only downward fluctuations of up to 90%. The latter look like partial eclipses because the burst decay remains visible in the residual emission. We revisit the interpretation of this phenomenon in the context of the new data set and find that it has not changed fundamentally despite the expanded data set. It is thought to be due to a disturbance of the accretion disk by outflowing matter and photons, causing obscuration and reflection due to Thompson scattering in an orbiting highly ionized cloud or structure above or below the disk. We discuss in detail the most pronounced burster SAX J1712.6-3739. One of the bursts from this source is unusual in that it lasts longer than 5600 s, but does not appear to be a superburst.