The specifics are going to depend on your implementation of Fate, but in Standard Fate (and Fate Core), when you Create an Advantage (or create a temporary aspect), you get a free invocation. I'm going to quote the rule from Fate Core:
Free invocations work like normal ones except in two ways: no fate
points are exchanged, and you can stack them with a normal invocation
for a better bonus. With a free invocation and paying a fate point on
the same aspect, a +2 bonus can become a +4, one reroll could become
two rerolls, or you can add +4 to another character’s roll or passive
opposition instead of
+2. You can also stack multiple free invocations together.
After you’ve used your free invocation, if the aspect in question is
still around, you can keep invoking it by spending fate points.
Via Jadasc, I checked Rob Donoghue's excellent post on Hard vs. Soft Tags. There's also a tangential reference to Out of Ammo on the Fate SRD. With this info, and a discussion on G+ to frame it, the following example comes to light.
When you put Out of Ammo on the target using the Create an Advantage move, the target is now out of ammo. There's no need for invocation, the aspect existing makes that a fact. (Nor, IMO, is there a need for a compel... you could compel the implicit Using a Gun aspect for a fate point to make it run dry instead of using Create Advantage to make the Out of Ammo aspect. But you specifically proposed using Create Advantage here.)
The target now needs to remove Out of Ammo and, if he's reasonably equipped with a spare clip, that would be an automatic action. Reloading, after all, is easy. It looks like your Create Advantage didn't do you any good...
It's THEN that the Out of Ammo could be free-invoked, to create a Fair (+2) difficulty where otherwise no difficulty would have existed. Now to reload, the target must do an Overcome.
Alternately, if reloading would already be difficult for some reason (say the gun in question is a revolver or we've already established that the target has no spare clips just loose rounds), so an Overcome against a difficulty was already going to be necessary. The free invocation could be used to make it 2 points more difficult.
You could use the free invocation as a bonus to your next defense claiming that the effort to reload cost him time.
There is also another alternative for creating the aspect that is overlooked. If a roll is missed, that doesn't necessarily mean failure. A failed roll can mean success with a cost, so you can also suggest the temporary aspect as a cost of success.
If a character has an aspect that relates to the limited ammunition in his weapon, then that could be tagged in order to make the character run out of ammunition. In that case, the exchange of fate points would occur as normal.
TL;DR - for your particular case, no invocation is required, the target is Out of Ammo. No Fate points are exchanged. He also doesn't get a fate point each turn he is Out of Ammo, though if anyone does compel the aspect while it exists for another effect that is not a Free Invocation, the Fate point exchange occurs as normal.