Typically when you're in space above a gravity well, you stay "up" by falling sideways so fast that you just keep circling around the object/planet. That's what it means to be in orbit.
In Star Wars, we rarely see ships in orbit. They usually just hover over a single place. Rather than moving sideways to stay up, they utilize Repulsorlifts to accomplish this feat, directly counteracting the effects of gravity wanting to pull them down.
When a ship is damaged or destroyed however, the repulsorlifts fail. Since there's no longer anything holding them up against gravity, they then begin to fall back towards the object/planet.
Now, regarding the discrepancy between large and small ships. If you launch a missile at a small ship, the power of the explosion compared to the size of the craft is more than enough to blow it into tiny pieces, wherein the force of the explosion launching those pieces every-which-way is more apparent than the pull of gravity for a time. If you were to take that same missile and launch it at a large ship however, it would do substantially less structural damage relative to its size (and impart less relative force), even if it is still capable of doing sufficient critical damage to disable it.
If a small ship were to take critical damage without being blown to tiny pieces, we should expect it to fall just as larger ships do. This just rarely happens.
Lastly, the Death Star was known as an orbital battle station, and it's almost always in orbit when we see it (Yavin, Endor, Jedha, Scarriff). It would make little sense for a craft the size of a small moon to use repulsorlifts, since it has no need to stop above a planet, or lower/raise itself relative to the surface (it's not like it needs to land). So when it explodes, gravity has no apparent effect, because it's acting equally on it the same as it always had been, continuing to fall sideways. The debris we do see falling and burning up are the parts blown towards the planet by the explosion itself.