I was thinking about this, 
wants 
to be his host, if Tryndamere did they the sword..Would the host body of Aatrox just stop doing anything or would Aatrox still have control of it for awhile after?
I always imagined that the weapon can choose to abandon its host, or compel them to die. It seems like a darkin can consume/destroy someone who is holding it, if it doesn't want them to... (Kayn color text etc)
Getting the person you WANT to pick you up, well that's a slightly different matter. Being able to whisper evil things into their mind probably helps! :-)
My guess is that the Darkin would still control the host, but if the host dies then the Darkin goes back into the weapon. So it would be in a Darkin's best interest to keep their weapon close to them in order to stave off host death for as long as possible.
We actually see what happens when a darkin drops his weapon in the aatrox teaser.
The host body will still remain and it looks like it can call the weapon back whenever it wants.
I guess that if someone else grabs the weapon while the host body is still active,then the weapon would simply try to corrupt the person while the host body would remain unaffected.(though I guess it could be harmful for the body if the weapon is too far away).
I'd imagine it depends on the duration of their bond and the state of the body and mind.
Someone like Aatrox, who has fully assimilated his current loan body, probably doesn't have that issue. Now, Rhaast on the other hand...
I feel as though a fully corrupted Darkin Host would be literally oozing with Darkin power. I feel as though the bond would probably be strong enough for him to pull off some boomerang action.
