from experience, a dealer will be more likely to fix something they "technically" don't have to for free if you've used them since buying the car from them.
At my work (indi) we regularly service cars that are still covered by manufacturers warranty, we do however use genuine parts when doing so.
We actually usually refuse to do any none manufacturer warranty jobs unless its one where the customer pays us, then its up to the customer to claim it back from their warranty company. This is due to the hours and hours we have spent on the phone to companies in the past while they make us jump through hoops and give the car's entire life history.
Most recently - Broken spring on rear of Mitsibushi Shogun 3 years old. "which spring was it, do you think they were towing, what caused the break, do you think it could have been a pothole etc etc etc. I hate them, trying to find a way out of paying out for a claim, ended up being very rude to the person on the phone and hanging up.