From what I understand, the Ascians travel to the various different shards and attempt to saturate each with a particular element - in the First's case, that element is light.
Once the elemental balance within a shard has been tilted far enough towards a particular side, that excess elemental aether starts trickling over into the Source, which causes appropriate, negative effects. Eventually, that trickle becomes a flood, causing the shard to crash into the source, resulting in a calamity and rejoining.
What I don't quite understand is how this relates to the Ascians' actions within the Source, or how the 7th calamity fit into this.
The 7th umbral calamity was caused solely by events taking place in the Source and didn't seem to have anything to do with any of the shards. So I'm a tad confused about how Bahamut breaking free and wrecking the place caused a rejoining, or how a rejoining caused Bahamut to break free (whichever is the case).