If you use a crossbow to tame these beasts, PACE YOUR SHOTS! The last two I tamed came within one arrow of death when they finally fell. The best way to tame them is to go to the desert and grab all the red crates, hoping for an ascendant longneck. Also, my perfect (semi)Universal trap is 5x3 foundations and ceilings, stone Dino gates on the 3 wide end, and thirty pillars, three high in the center of each border foundation on the long side.
Just build a 2x4 or 2x6 pen with a dino gate on one side, placing stone pillars on the outer edges all along it. Make it near the green obelisk! Then fly the griffin into it thru the door with a pteranodon, fly out thru other end thru the pillars, land your mount, run to the other side, close the door, boom. Trapped griffin. Then tame with roughly 150 tranq arrows. Use mutton or prime meat as they are easiest to come by.