Without mentioning the brand, it is very difficult to give you a direct answer. When you say European, it can goes from Rolls Royce to a Volkswagen, and everything in between. Some are more generous with their warranty and some are not. Usually, for a car that is a couple years old, assuming that it did not exceed the mileage portion of the warranty, there should still be some original manufacturer warranty remaining. For most brands, these warranties are fully transferable to the new owner. I believe for certain manufacture like BMW and Mercedes, if the cars are resold at their own dealers, they can become a "Certified Pre-owned" where they get an extended warranty on the power train. However, there is usually a $1000-2000 premium on these "Certified Pre-owned" as compare to non “Certified Pre-owned.”
I’m sure you can “ask” the GM dealer to cover the cost as part of the agreement, but no one does things for free, and they may charge you for this “service”. What I would recommend you do is to call up the manufacturer, and explain the situation; they should be able to give you a better answer.