If an advanced race builds a Dyson Swarm comprised of millions of habitats and billions of solar collectors orbiting a star, and then are completely and inexplicably all killed with minimal damage to what they built, what would happen to their Dyson Swarm? Assume the swarm is located 0.3 to 1 AU from their star, a yellow dwarf like our Sun. I would think that without course corrections being made, the various parts of the swarm would collide with each other over time, scattering fragments everywhere and causing more collisions. Would the objects tend to fly off into space, or would their orbits decay into their sun? Over millions or billions of years, would the fragments aggregate into a new planet or something? Would they form a new asteroid belt of sorts, but in a sort of thick shell around the sun, instead of in a ring, due to the high orbital inclination of many of the objects in question?
I would think that without course corrections being made, the various parts of the swarm would collide with each other over time
This suggests that not only did the builders poof out of existence, but any automated systems they constructed poofed with them. I can speak only for myself, but I like things that look after themselves rather than requiring me to babysit them all the time, so any dyson swarms I make certainly won't immediately fall apart in my absense.
The ability to construct a dyson swarm doesn't necessarily require the ability to make intelligent and/or self-replicating systems, but it certainly implies it, and such systems might be self-maintaining. Conversely, there are arguments that complex systems that demand their users actively maintain them ensure that critical skills and knowledge can't be lost, so I guess it is up to you what sort of situation you're dealing with here.
As to whether the elements of the swarm would collide... well, that depends very much on the density of the swarm and the nature of their orbits. There's a lot of room in a .3 to 1.0 AU sphere, so there's no need for everyone to be elbow-to-elbow. The ultimate long-term stability probably depends on external gravitational influences, but if you've dismantled a generous chunk of your stellar system to make your swarm you might not have to worry about that too much. Eventually you might have to worry about close encounters with other stars, so you probably couldn't make a dumb system that would remain intact in perpetuity
Would the objects tend to fly off into space, or would their orbits decay into their sun?
It takes a surprising amount of energy to do either (from earth's orbit, it is energetically easier to fire you into interstellar space than into the sun, as it happens) and in the absense of significant energy input from somewhere outside the swarm this seems unlikely.
A swarm made from statites rather than regular boring orbital elements might be capable of partially blowing away into interstellar space if the payload part of the statite were subjected to sufficient mass loss some how (such as in a collision). Severed payloads (or the debris from broken ones) would then fall inwards and might well drop into the star in due course (not sure how much energy you'd need to add to drop the debris into a very eccentric orbit instead of a death-dive, nor how stable that orbit might be, but it could happen).
Note that constructing a dyson swarm is probably best done after you've taken care of problematic asteroids and comets which might risk it, so really high energy collisions are probably hard to come by in the near-to-medium term, but it is possible that long term gravitational effects from any big planets in your outer solar system or passing stars might kick a load of new comets into the inner solar system, and those could certainly add a lot of energy and smash everything up good.
Mostly though you'll probably just find your lovely neat orbits get messed up a bit, and the ring will get smeared out and clumpy. Maybe given enough time the clumps could smoosh together and form a denser asteroid belt or new planet, but there are some tricky questions about density and planetary formation there which don't seem to have any nice easy answers.
Would they form a new asteroid belt of sorts, but in a sort of thick shell around the sun, instead of in a ring, due to the high orbital inclination of many of the objects in question?
Well, the swarm kinda already is a new asteroid belt of sorts, so it won't really form a new new belt, just a messy version of the old new one. If you see what I mean.
If they are orbiting the star, and there is no energy supply, they cannot escape the gravity well. Thus they will either fall into it, or coalesce into a big space clump like an asteroid or a planetoid as a consequence of momentum dissipation following collisions.
More likely a part will follow one destiny, the remaining part the other.
If instead there is an external supply of energy, like the gravitational swing of a passing by body, some of them can also be sling shot into longer orbits, some even escaping the system.
In any case progressive and cumulative damage will make the swarm unusable.