When you destroy a character (Pawn class) the controller un-possesses it. When a new pawn is created the controller needs to possess it.
When a Player is killed you should re-use the character. Go through all the normal ragdoll, movement logic, un-possess etc, then teleport it to the respawn location.
On respawn, re-enabled the character and then possess it.
Possession events need to happen in the Game Mode.