I don't have too much to add since Anonymoose did such a great job in his post, but I can give a few of my own thoughts on the matter. First off, the goal of the Black Robed Ascians seems to be bringing back Zodiark, mainly by causing chaos. They do want rejoinings, and would rather avoid floods. Igeyohrm, for example, is mentioned in the Encyclopedia Eorzea as being shamed and/or punished for letting her shard get flooded by darkness instead of being rejoined. Our white robed friend appears to work hard to prevent those things from happening, sending Unukalhai to prevent the Warring Triad from draining Eorzea being a perfect example. Each Ascian seems to have their own pet projects and preferences as well. Lahabrea (despite being titled 'The Warrior') tended to favor manipulating mortals, the Minor Ascian Order in the Summoner Questline sought to cause chaos through magic (specifically necromancy), Nabriales preferred direct force, while Solus created the Garlean Empire. This means they can sometimes work at seemingly cross purposes, and the ire which Solus had for Lahabrea indicates they don't always get along even if they're united in goal. This also makes them ideal MMO villains since they have general/vague enough goals to be involved in nearly any conflict. That being said, there is something else going on as well....
Elidibus (who's practically our 'main' Ascian at the moment) seems to have different goals, and especially different aims, from the other Ascians. The rest of them seem to regard him with some caution, even if they agree he is 'good at his job'. Solus's wary nature towards him is the clearest evidence of this, but there are even bigger ones. Elidibus is implied to have specifically taught the Scions how to kill Ascians after all, through teaching the Sahaguin priest how to use the Echo to use Ascian style immortality. The fact that Nabriales immediately moves into kill mode after realizing this shows how the others probably weren't in on that particular plan. He also claims that if the Scions master the echo enough, they will 'be on the same side' as him, whatever what means. My personal pet theory is that he actually wants Hydelyn and Zodiark to fuse again and create 'true balance', even if it would logically wipe out all mortal life since Hydelyn is the thing that creates/sustains it. Either way, he is a man of mystery, and his plans seem to ebb and shift around. Now he's in possession of an especially powerful body both physically and politically. Who knows what his plan is?