We define "Slow LDEs" as soft X-ray flare events with gradual rise phases, as well as long decay times. Celebrated examples of such events include February 21, 1992 (the Yohkoh "candle flame" cusp event) and August 28, 1992 (a prototype coronal dimming event). We have found another striking example in an event of January 20, 1999, which exhibits a fan-like X-ray structure above the post-flare loop system, as also observed in the Augest 28, 1992 event. Such events may seem thermal in their nature, since by definition they have no impulsive phase. We confirm, however, that such events invariably exhibit hard X-ray bremsstrahlung emission, a signature of non-thermality. The slow LDEs also follow the pattern of the Neupert effect. We have surveyed the Yohkoh data for events of this type, and find several examples of the recently discovered supra-arcade downflows (McKenzie and Hudson, ApJ 519, L93, 1999). The open questions are (i) is the "slow" hard X-ray emission physically different from that of the traditional impulsive phase, and (ii) is it related to the observed velocity field? Current hard X-ray imaging instruments have been unable thus far to image the weak coronal hard X-ray component associated with events of these types, but HESSI may be able to do so.