Build a stone 5 by 5 1 stone Dino gate tthen with pillars build to the top of the gate then fill top with ceilings and put a door on the gate frame. Then run through get the griffin in and then run around and close the gate. I suggest using a rifle since tranq arrows will possibly kill it
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.