You need to track down four hot air balloons and take them down. Work your way up to high ground and track down the first balloon. When you get close enough, take out the UAV circling it (watch out for bombs from the UAV and Shockwave them away). One or two Lightning Bolts should deal with the UAV.
Then, call down a Lightning Storm on the balloon itself to take out the shield (you don't have to hold the Storm for a long time). Finally, work your way close to the balloon, jump over to the platform at the bottom and walk around decoupling three fasteners (press 
when you get the prompt). You will probably come under fire while you're doing this. If it gets heavy, take them out with Precision. Be careful using Lightning Bolts, Hammers or Grenades around the unshielded balloon. You can easily blow it up and have to start over again.
You'll hit the ground as the platform comes off and you'll be under attack from suddenly hostile civilians (as well as surviving gang members). Get up on the rooftops again to avoid the gassed civvies and track down the second balloon in the Warren. Repeat your earlier performance.
Neon balloons[edit]
Now it's time to take out two balloons in the Neon. These balloons are tougher. First, you'll be under constant attack from civilians (even the police are shooting at you!). Second, the balloons rarely pass by buildings high enough to allow you to reach the hanging platform and their shield recharges in about a minute, so you have to move fast once it's down. Third, just to make things even more interesting, the platforms mount machine guns.
Approach the balloons carefully and try to find a high point along its path. You'll have to take out the shield with Lightning Storm and then use Precision to destroy the machine guns. If you're fast enough, you may be able to reach the balloon before the shield comes back up. More likely, you'll have to take down the shield once to take out the machine guns, and then take it out again to get to the platform. Once both Neon balloons are down, the mission is over.