I want to replace a standard duplex receptacle in my kitchen with one that provides integrated USB ports. The existing receptacle is protected by a GFCI receptacle upstream in the circuit. Is this against the NEC? Will I encounter any problems with this setup?
There is no fundamental reason why you should not be able use an integrated USB outlet on a GFCI circuit. There should be no stray current going elsewhere that would trigger the GFCI, and the 5V USB output is isolated from the mains current anyway.
Actually there is a possibility of leakage current that may trip. These USB sockets and even standard USB chargers actually leak current. As long as your national electrical code applies to IEC60950 which most do and depending on its Class rating it should be fine.
It is not unknown for people to have used genuine iPhone devices connected to genuine iPhone chargers or any other tablet/phone devices and to have up to 90V between the case of the device and ground. This effect if using the device can feel like static shock or tingling.
Most people would think it a fault and or dangerous however, on the contrary this is not dangerous though and the current should not exceed 250 micro amps. Generally these chargers have one ceramic cap connected from the primary (mains voltage) side to the output. the purpose of this capacitor is to reduce the conducted electromagnetic interference (EMI) so that the power supply can legally be sold and meet the legal requirements that involve the leakage current being low enough not to cause a hazard to a person that touches the output connector of the power supply.The leakage current can legally be high enough to give you a real shock if your hands are wet. but not high enough to stop your heart. The unfortunate thing is that these power supplies that do not have any ground connection on the primary side will leak some current through this EMI capacitor, so if you measure the voltage between the output and earth ground with a high impedance voltmeter you will see a significant ac voltage. That is just normal.
The ultimate point been depending on the sensitivity of your GFCI/RCD circuit it could trip.