For an easy tame, wait for the Lymantria to land and then place a tent directly where he/she is at. Its stuck until the tents durability dies out, this gives you enough to build a little house around it for a safe, easy tame.
If you cannot make a tent you can bola it, then place a wooden pillar basically in the middle of it. Then you put a thatch roof on the pillar and surround it by thatch door frames. It will be trapped and you can shoot it without it getting away.
The Lymantria is like the Pteranodon on The Island, but it takes way more torpor then the Pteranodon. So easy taming is bola them (or wait until its landing) and place a tent on it ( DONT place the door of the tent in line to the head of the moth! It can really easy escape then. I hope I helped some of u Guys. Sry for my bad English btw.
Best place to tame them is the Dragon cave at Floating Islands on Crystal Isles. Just tamed 3 150s stuck in the nooks and crannys. Bring bolas to ensure the snare. Will always break out of the first bola regardless of what tranqs you use (best results come from darts as they're more delicate than arrows) so be prepared to either shoot it again if it gets stuck or bola it if it starts to flee!
I tamed the lymantria in early game valguero and it was actually quite easy. All you need to do is use a bola (I used a normal one) and then keep attacking it with a club and it should get knocked out in couple of hits. However, if your trying to tame a higher level one, bring multiple bolas. The saddle is quite cheap so it can be utilised early game.