This is a theory, and it is worth noting that I have only played Skyrim (and it's DLCs), Oblivion (with Shivering Isles and Knights of the Nine), Online, Legends and Daggerfall. Most of my resources come from the Wiki alone.
I do not think that something killed the Dwemer. Instead, I believe that the Dwemer were able to trap their souls into gems, and embody their Animunculi. Most Dwarven contraptions have soul gems upon looting, ranging from Petty quality to Grand. Of course, you need Black soul gems to entrap Men and Mer (Such as Bandits or merchants), so my theory may have just been disproven, if it weren't for the Falmers' in Skyrim having souls which can be trapped with white soul gems.
If the Falmer, a race of elves, could be trapped using white soul gems, then the Dwemer may have been able to put their souls into white gems too. This is supported with the fact that you see no Dwemer in the Soul Cairn, which only houses souls trapped into Black soul gems.
If the Dwemer did have Black quality souls, they may have used a ritual similar to what you witness in the Mages Guild questline in TES IV: Oblivion, where a Worm Cultist transformed a Grand Soul Gem into a Black one, except doing the opposite. They could have stored their souls into the Black gems, performed the ritual and converted the filled Black gems into the white ones.
This is all speculation of course, and any other points of view would be greatly appreciated, and further inspected on by me.