I think the connecting green lines is the charged missiles ability of the Assailant. It allows their phase missiles to do some degree of splash damage.

Vasari are tough because of their support cruisers: Overseers and Subverters. Subverters are especially bad because they have an area effect stunning ability that can only be described as stupidly powerful. To defeat Vasari, you have to take the support cruisers out of the equation. Cobalts have an ability disabling attack called Sabotage reactor that can help. Also, the Dunov battlecruiser's EMP charge removes anti matter, plus it's area effect. Against an AI, the Dunov solution works well, but a human player will be wise to that and take down your Dunov first. Cobalts are difficult to use by that point in the game because they will be destroyed in prodigious numbers by Assailants.