Kay so I just caught A bulbori but it didn't show up in my stables or anything and now I'm out a trap and experience for my pets D:

Is this like something that happens all the time or what?

Sylesties have a chance of running away. It should say in the combat log.

You sure you caught it and it didn't flee when you threw the trap?

Nope It said The bulbori was caught and my team was victorious :/

If a Sylesti is the only enemy left in the battle and you are attempting to catch it... if you catch it or if it runs away, the battle will end, your party will be victorious, and your pets will receive Experience for any pets defeated (catching or running away does not count).

As stated, Sylesties can run away when attempting to catch them. There is no visual difference between them running away or being caught other than if they are caught, it says they're caught in the battle log and the Sylesti is now in your Stable. If it runs away, it just simply says it ran away.

To my knowledge, this feature is working properly.

If this happens again and you can't find your caught pet, can you take a screenshot of the battle log saying that the pet was captured?

If it happens again I'll most definitely take a screenshot