How to Destroy the APC
The Falcon Lost Incursion is 15 waves of enemies, mostly made up of elites (purple health bars), with a few exotics (yellow health bars) spread throughout and coming in later on. On waves 4, 8, 11, and 15 an exotic enemy will spawn with explosives. After thinning out the wave enough to take out the marked enemy, your task is to pick up the explosives and run them to the back of the APC.
In order to survive the trip, two other teammates will have to turn off the automated turrets on either side of the armored vehicle. To the right, the panel is on the far side of the room, just around the corner from the APC. Behind the staircase and under the balcony, you can find it secured to a metal support beam. To the left, the panel is on the right wall of a small room directly next to the APC. Unfortunately, each turret will restart itself within a matter of seconds, meaning both must be turned off at the same time, just as the explosives carrier runs behind to plant the bomb.
If you wait too long before planting the explosives, more enemies will continue to spawn. So aim to have the wave at a manageable level instead of trying to clear out the entire wave before making your move.
When the bomb goes off, the APC will begin a missile barrage, firing sporadically around the room. Stay on the move and get back to a safe distance before hunkering down for another few waves. Planting the forth set of explosives on wave 15 will trigger a similar missile barrage. Survive it, and the APC will explode, giving you one complete incursion.