In order for the fragments not to simply reform into a single body from their own gravity, the damaged moon would have to meet certain criteria.
Either it would have to be within the planet's Roche limit, or the fragments would have to have been launched at above the moon's escape velocity. Since you want a slowly evolving situation, the latter can be ruled out, so we need to assume the moon in question is so close to its primary that it was on the verge of tidal destruction anyway -- and when the "damage" (pretty much have to be an impact) took place, the fragements had tidal forces separating them with barely more force than the gravity that would tend to bring them together.
If the timing was just right the fragments of the moon would drift apart over a period of months, years, perhaps even generations. The smaller bits that were blasted clear by the impact would already have begun to form a ring system, though it would be faint at first -- but as the major fragments, already fractured by the main impact, begin to tidally separate the collisions between them would increase the count of small fragments, filling in a ring (more like those of Jupiter or Neptune, rather than the huge gaudy system Saturn shows).
The effects of all this on the primary would be, first, a heavy meteorite bombardment near the moon's orbital plane (likely the planet's equator as well); this will continue for a long time, as fragments decay orbit. Second, there would be a huge change in the planetary tides -- assuming the moon was large enough to raise noticeable tides in the first place, those tides would gradually lessen as the peaks and valleys of the cycles spread out, eventually vanishing entirely as the moon finishes its transformation into a ring.
As noted by Morris The Cat, these same tidal effect changes would also apply to the primary's crust -- though in the case of a moon already near its Roche limit and a breakup event "barely big enough", these would be gradual rather than catastrophic. You'll still have plenty of trouble with the impact belt (and it'll cause some tsunamis, too); it might still be an extinction event, but the required breakup impact is much smaller in this case and it need not be an instantaeous catastrophe.