Hey guys! I know this is an old post but I just got back into editing again after a very long break and saw an unanswered question in the comments that I was experiencing too.
The Blink Strike will trigger the effect of blinking, damage, and whatever effects you may put in, but the cooldown wouldn't go off. Essentially you could cast it as much as you wanted without a cooldown as long as you had the mana.
I believe the issue behind this was that there are a lot of wind up animations for certain abilities and on certain models. For example, the Mountain King winds his arm back before he throws a Storm Bolt. However (continuing with the Mountain King & Storm Bolt example) the "A unit Starts the effect of an ability" begins its Event Trigger as he starts his cast animation, and the cooldown of Storm Bolt does not begin until the hammer leaves his hand. This means the ability technically isn't used, doesn't go on cooldown, but the trigger registers it as an ability cast so it does the remainder of the trigger as if you had.
This is a short fix I used for it, it may not be the cleanest way of doing it but it did the job.
I did 3 changes here compared to the original that made the difference.
1. I changed the "A unit Starts the effect of an ability" to "A unit Begins channeling an ability". This ties in with the second change I made.
2. I made the base ability a channeling ability. Since the ability I wanted to Blink Strike with needed to be a target-lock on, I used Siphon Mana. It's a channel that targets a specific unit, perfect for the uses.
3. I put a wait timer of 0.25 seconds. This gives the model a chance to go through the cast animation before the trigger kicks in. Since 0.25 seconds is a very short time, not even one tick of the Siphon Mana goes off on the enemy, but since you started the channel the cooldown of Siphon Mana will begin. When your trigger kicks in 0.25 seconds later, the "Unit - Order (Triggering unit) to Attack (Target unit of ability being cast)" action will instantly cancel the channel of Siphon Mana, giving you the Blink Strike ability you desire *with* a cooldown.
If you want to do this as a free target ability (*Not* for a single target lock-on) then I recommend Blizzard as a base spell.
Also the other change I made (Not Necessary!) is that I turned the single target damage into an AoE damage upon landing. That has nothing to do with the changes needed whatsoever to make the Blink Strike work with a cooldown.
Final note, just remove your "Stats - Buffs", "Sound - Effect Sound (Looping)" and "Art - Lighting Effects" from your Siphon Mana ability so there's no chance of visuals or audios from your Siphon Mana apprearing, and re-do the numbers on your Siphon Mana so that you're draining health instead of mana. This allows you to target enemies who don't have mana bars.