The implants become encapsulated if they are made of silicone and that tight feeling when you animate is likely the lower facial muscles pulling on that capsule. It is generally nothing to worry about. At one year, the implant is very stable.
Facial implants usually fixed with screws. Even Jaw, chin or cheek implants fixed there is a minor chance of displacing during performing martial arts. My suggestion is waiting until all soft tissue covers the implant. It is really hard to give any time for this but around 3 months time all soft...
The labiomental fold is a direct reflection of the projection of the chin. The problem is not lacking tissue in the fold it is typically excessive projection of the mandible.
Best,
Mats Hagstrom MD