What to do with dislodged asbestos?
I recently abated my house's asbestos exterior siding and did a LOT of research before doing so.

The short answer is, don't vacuum it. Use something wet to pick it up (like mopping or a carpet shampoo machine) instead. Mostly, you'll be fine.

I called my municipal inspector, state regulator, and the national advisory council and they all gave me the same answers. They compared the risks of asbestos with the risks of cigarettes. With cigarettes, any given cigarette could kill you. More cigarettes means more opportunities to find "the one" that does.

Asbestos is NOT like that. Asbestos does, in fact, damage your lungs anytime you breathe some, but it's the same kind of damage as typical dust. The problem is ACCUMULATION, not probability.

For more detail, there are three types of asbestos. The smallest kind is too small to do any damage. The large kind is big enough that your body deals with it very well. The medium kind causes problems. So that's most of the asbestos that's not a problem. The remainder will do damage, but it's nothing you'd even be able to measure. The problem is when that damage accumulates from many many exposures over a period of decades.

They gave me 3 pieces of advice.

Firstly, it's not dangerous if you can't breathe it. In most cases, this means it's safer to leave it alone than to try an abate it. In my case, this meant trying not to break the tiles any more than necessary.

Secondly, wearing a mask is really effective. I used a typical dust mask that filtered pollen. You can find more official recommendations for what is rated for filtering asbestos.

Lastly, keeping it wet will keep it out of the air. This relates to your vacuum idea. Using a vacuum will make it easier to breathe the stuff.

My recommendation would be to spray the floor with soapy water to keep it from going up in the air (carpet or hard floor). If it's on a hard floor, just mop it up, dig a hole and bury the sludge that you mopped up. You might throw out the mop, too. If it's on a carpet, I'd use a shop vac to suck up the water. Water, vacuum, repeat until you feel like you got it all. Bury the water if it makes you feel good, but even just throwing it out isn't going to hurt anyone.

Now, there's an important caveat... you need to check the local regulations. Neither the Federal government, nor the state of Texas have any regulations about a homeowner abating their own asbestos. None at all. The only requirements were when I took it to the dump. They just wanted it in marked trash-bags... then it just got dumped in with all the rest of the trash.

Your jurisdiction might have regulations about how you dispose of asbestos as a private homeowner.

The important thing to remember is that it's the lawyers who have made it scary. MOST governments don't have any regulations at all for private owners taking care of their own home. But it's important for you to do your own research. I've done a lot, but I'm not ACTUALLY an expert.