SC2Mapster

You want an Effect - Target type ability that uses a Create Persistent effect. Under the ability the Smart Command flag allows the ability to be linked directly to the mouse click. Use Modify Unit effects as the initial and final effects of the Create Persistent effect to turn the unit/turret to the target. You then need to have a single period on the Create Persistent effect that uses your Launch Missile effect at an offset. Under the Launch Missile effect the Effect: Periodic Effect can be used for collision detection.

You can use the No Markers validator to stop that since the damage effect applies marker to the target units. Units like the hellion and colossus also use that validator to stop attacking the same targets several times.

If that is not what you want then what do you want for "I would like to know how to keep firing toward and tracking the mouse cursor"?

Thought you wanted it to pierce foes. If you want it to destroy itself on contact you use a Set effect for the periodic of the missile that uses two damage effects. The first hits the target and the second has the Kill flag and targets the source (the missile).

If you set the Create Persistent that it uses the source unit as a starting point it is. Check the knockback ability of Alarak as an example. 

So turn the ability on and then it constantly fires at the cursor? You probably will need triggers for that. Use a Behavior type ability to apply a Buff behaviour that the trigger event recognises. This trigger then turns on another trigger that periodically uses the effect towards the x/y coordinates of the mouse.

You will need a second Action actor linked to the Damage effect for creating the impact visuals.

If you do not want to see the original impact model then remove it from the Action actor.

Some of those abilities apply a buff on the target units that prevent other searches targeting the unit using a Unit Compare Behaviour Count validator.

The set effect has the validator so it is not applied to units with the buff. The buff itself has a low duration but is enough to prevent the unit being hit multiple times. You could even make the validator require the caster effect history and make the buff stack allowing multiple units to attack at once.