Yes the g/tee stays with the car.  When we bought ours second hand, I
contacted Hyundai UK with the cars reg and vin and they sent me a new owner
sticker to put in the handbook / service booklet.
Gio

Here in the USA a used car purchase is 2 ways.  From the original owner
directly, then you contact Hyundai to inform them you are the own of
the VI# & they pass the warranty onto you with the roadside assistance
that is for 5 years from date of original purchase that will also
follow you.  Then you may be purchasing it from a dealer, then that
dealer put whatever warranty on it as his "certified cars are covered".

Here in the USA a used car purchase is 2 ways.  From the original owner
directly, then you contact Hyundai to inform them you are the own of
the VI# & they pass the warranty onto you with the roadside assistance
that is for 5 years from date of original purchase that will also
follow you.  Then you may be purchasing it from a dealer, then that
dealer put whatever warranty on it as his "certified cars are covered".

You don't need to take any action to transfer the 5/60 warranty.  It stays
with the vehicle.