Depending on the type of implant and the surgeon's preference, training, and habit, there are different ways of trying to ensure the chin implant doesn't "move around" or become malpositioned during healing.
I use extended anatomic Flowers mandibular glove vertical tilt solid silicone elastomer chin implants the most, as these provide a secure anatomic "fit" with the normal bony contours of the anterior mandible. They almost "snap" into place, but the pocket must be carefully created to be the exact size and position you want the implant to end up in. Exact pocket size is the best way to keep an implant immobile, but there are also permanent or dissolving sutures to hold it in place as the scar heals around it and then it's pretty much "stuck in place!"
Medpore implants have tiny interstices ("pores") into which tissue grows, immobilizing it. Of course, this needs to be kept in proper position until that tissue in-grows, so exact pocket creation again is important, as well as a temporary (or permanent) fixation suture.
Bony chin surgery (sliding genioplasty) involves sawing off the underside of the chin and then advancing it into the desired position and then screwing it into place with titanium plates and screws. This sounds worse than it is, but frankly, though I have had significant craniofacial training and am fully capable of performing this operation, I think it has a proper place in only a minority of patients. The anatomic silicone implants are just that superior, IMHO. They can be easily removed or changed in size also (though this is rarely necessary if your initial choices are good), whereas the bone surgery pretty much is what it is and cannot be easily revised.
Also, while the "invisible" intra-oral incision may seem to be a good way to go, this incision choice has a higher risk of infection than the submental incision (just beneath the chin, where many of us already have a scar from that clonk on our parents' coffee table as a toddler). I prefer the external approach because of the lower infection risk, the better visibility of the mental nerves (these should be preserved to avoid numbness of the lips, gum, and chin), and the fact that, properly closed, it heals nearly imperceptibly. This also allow me to remove the fatty pad under the chin that improves the cosmetic result by an additional factor!
So as you see, it's less about what is used, but rather how carefully the surgery is performed that keeps the implant where you want it. If it "slides around" after surgery, there is no chin strap or elastic chin holder that will secure the implant as well as better surgical technique! For examples of my patients undergoing this procedure, click on the link below. Best wishes!