The Core is along the precarious walkway. However, someone grabs the Core just after the Kid arrives, making the trip even more precarious.
After someone grabs the Core, the region will collapse partially. Certain floor tiles will fall a few seconds after you step on them. The initial segment only has a few enemies, nothing out of the ordinary from what you have seen. When you reach the junction, the path to the right has some Fragments guarded by two Security Turrets, but has a collapsing floor throughout the ring. To proceed west, you need to destroy the turret in front of the gate.
The path leading to the next section has a few special turrets. If the projectiles are reflected, they may still try hitting the Kid once again, but at least they're slow-moving. When you reach the Skybridges, there is a Bronze Spyglass memento reached by walking along the small path to the platform. The following Skybridges lead to minor enemies that try attacking the Kid, before finishing with a large Scumbag. Defeat the large enemy to open the gate.
The next path contains a Hand Grenade, which can hit targets when you throw it over a wall. It easily destroys the two groups of enemies that pop in. Additionally, you can use the nearby Skybridge to obtain Something Burnt to upgrade one of your future weapons. Head to the Skyway to complete the level, although there are a few Gasfellas that are sleeping (and will awaken if you destroy some of the boxes.)