Now you can continue along the path in that direction, but blue flames will burst out of holes in the wall to the right. Knock them out of the air, attach the handle to the nearby wall, then have Hagrid give it a mighty tug to cause the back wall to drop down. Just past the dragon cage, along a back wall, some gremlins hold a lever handle aloft. It'll abandon its 'torch the path ahead of my cage' strategy and instead focus on the newborn, which allows you to conveniently pass that particular hazard and proceed toward the right. The egg will drop into a new basket and hatch, which of course inspires the large dragon to feel motherly. Lift it into the second pipe, which is also ringed, then carry it through to the far end. Enter the cage and approach the egg, then use your magic to lift it up and out of the first set of tubing.
Blast the lock on that cage's door with your magic, which will cause said door to swing open. It'll roll through that device and arrive on the floor in the cage behind. You can use your magic to lift it into the series of piping with colored bands.
In front of the left cage, there's a basket with several eggs in it. Give the dragon something else to think about.