Most fillers are made of hyaluronic acid, which is present in the body naturally (mostly in skin, but also in cartilage and vitreous humor). Long-lasting fillers may add an additional chemical to stabilize the HA and slow its breakdown. HA is degraded locally in the skin by naturally occurring hyaluronidase enzymes and by free radical degradation. Once it's broken down into small enough molecules, it enters the bloodstream and is excreted via the kidneys and liver. Note that the breakdown process is identical to what's happening in normal skin as it maintains itself so it's not like it's taxing or damaging to the kidneys or liver.
As for long term use, yeah, the stuff dissolves but the actual physical act of injection and local trauma to the skin provokes an inflammatory response which eventually results in collagen deposition (scar). So even after the stuff dissolves you have a permanently slightly plumped area because of the scar. You do these injections regularly for years and that scar will get bigger. Sometimes that's a good thing, you can get by on less frequent injections. (Sometimes, if there is an exuberant inflammatory reaction, people get big lumps, which can often only be treated with steroid injections or surgical removal.) So I don't think the lips would ever get "saggy" per se if someone stopped using fillers, because of the scar.