Based on how there's a cable linking the transporter gate to the destination, we can probably guess at how they work. Garleans can't manipulate aether but they are still comprised of aether like everyone else. If a Pure-Blooded Garlean were to try to use an aetheryte, best case scenario is nothing would happen, worst case is their aether would dissipate and they'd die. I posit that the transporter gates work by breaking you down into aether and then transferring you safely through the cable, then reconstructing you on the other side.

Regula's teleportation was likely a wireless version, connected via a pair of beacons within his armor and the Gration.