The various extracts regarding the False Desolation gives me the impression that Voidlight was directly distributed by Odium via Connection from the Spiritual Realm, then during the False Desolation, Ba-Ado-Mishram somehow started doing the same.
I know the main idea behind the question is whether or not the Everstorm distributes Voidlight like the Highstorm distributes Stormlight. In Oathbringer, spheres left out during the Everstorm were not filled with Voidlight and from what we've seen in the Rhythm of War preview chapters, the Fused apparently need to withdraw to refuel. I posed this question on the Discord once, were Gavilar's black spheres actually like Rosharan spheres at all? Does Voidlight need a different storage medium than Stormlight?