After reaching the new location, hear out what Ra's al Ghul has to say. Follow his instructions, approach the chalice #1 and press A to drink the Blood of the Demon. Wait for all the walls to disappear and move a few steps forward #2. Listen out to the details of the upcoming trials and you will learn that you need to reach the place marked by al Ghul without touching the ground or any other elements of the environment.
Thoroughly read the instruction displayed on the screen. Approach the ledge and note the blue trail left by Ra's al Ghul, suggesting a route you should follow. Jump off the ledge and start gliding. You will need to alternate between doing two things. The first is the Dive Bomb, performed by holding down the right trigger #1 and the second is Pulling Up done by letting go of the right trigger and at the same time leaning the left analog stick down #2.
Don't worry if you won't get the drift and fail a couple times, as some training is required here. What's important is trying to perform the two mentioned moves so that you stay as close to the blue trail as possible. Your goal for now is reaching the blue crystal platform #1. It's worth mentioning that in case you have lowered the altitude too much, you can always use the Grapnel Gun without any penalties. After reaching the platform, hear out what Ra's al Ghul has to say #2.
The next trial is very similar to the previous one, though this time you will also have to dodge obstacles on your way #1 and correct the direction you're flying from to time by turning left. Keep in mind to always fly following the trail left by Ra's al Ghul. In this case as well you're objective is reaching the crystal platform, on which you can land both conventionally as well as using the Grapnel Gun #2.
A couple assassins will appear on the platform you've landed on #1. You need to use the same tactic as with other enemies equipped witch blades, so often evade or jump over your foes #2. Luckily the assassins aren't very tough. After you're done with them, Ra's al Ghul will show you the further flight route.
After jumping off the ledge you will of course need to alternate between diving and pulling up, though this sequence differs from the last one in that you don't need to reach a platform, instead having to fly to a large vortex #1. Fly directly above the vortex and hold down the right trigger to dive into it #2.
If you've done everything properly, you safely fly through the vortex and reach a new area, landing on top of a small tower #1. Listen to Ra's al Ghul's speech and jump off the ledge. The main difficulty in this trial is the need to fly through the elements of a steel construction. In result you have to pay more attention to choosing the moment in which you pull up #2, as that's when you have the least control over Batman.
Right after flying through the steel construction you should see a new crystal platform in the distance #1. Just like before, you can either land on it or use the Grapnel Gun. Hear out what Ra's al Ghul has to say and continue on. In this particular case the whole diving and pulling up segment will last a bit longer #2, though you won't bump into any obstacles on your way. Your objective is reaching the new crystal platform.
Once again be prepared to fight assassins #1. You have to stay especially cautious, as this time they will be in a bigger group. The overall tactic stays the same, though you might try to throw some of the enemies down from the platform #2, which should make winning easier.
Start the last flight. This time you will have to reach another vortex #1 and just like before dive (by holding the right trigger) in the right moment #2.
Note that you've returned to the same room to which Talia has brought you before. Get out and go up the stairs #1, returning to the Chamber of the Demon. After reaching the gate, talk with Talia al Ghul #2. You will have to watch a longer cutscene, during which Batman will refuse to take the new position. It will lead to a fight with Ra's al Ghul.