Proceed along the path to come across Squirts, Gasfellas, and Security Turrets whose projectiles ricochet. Eventually, you'll be able to activate a switch that raises the cage holding the Scrap Musket, a gun with more emphasis on spread than range. Use it to eradicate the Squirts that spawn, and then to destroy the Squirt generators up the stairs. Up another set of stairs is the Arsenal, where you can swap weapons if you desire. Destroy the Security Turrets in the area to open the gate, which will allow you to retrieve the Core.
Past another gate are Gasfellas that swing their axes around them in a vicious circle. Keep your distance and fire away at them, as they lack movement while swinging. Up another set of stairs are more Gasfellas and some Squirts that pop out of crates. Try to destroy the crates before they can open to avoid having to fight the Squirts completely. Along the floating path ahead, Security Turrets will spawn. Their distance proves to be an obstacle for the Scrap Musket, so make sure to counter-block their shots if you have it equipped as your only ranged weapon to deal extra damage. Past these are some Stabweed plants that can easily be shot out of the way with the Scrap Musket, and some more Gasfellas.
Past some flamethrower Security Turrets is the final set of steps, which lead to an arena. Fight off the many enemies that spawn to be rewarded with many Fragments that drop from the sky, and a Marshal's Badge memento that appears in front of the exit. Take the stairs up to the Skyway to return to the Bastion. Upon returning, the Stranger will draw your attention to a pipe, which the Kid can interact with to enter a Dream.