Ideally you want the HDMI format to change to match the content being played. So for your streaming sources like Netflix it should switch to 23.976 for most of their programming. Playing a 59.940 source at 60.0 would produce a repeated frame every 16 or so seconds. Some people are bothered by this while some are not. Best place to notice it would be things like smooth camera pans or scrolling news tickers at the bottom of some stations.
Once you enable TV motion settings that do processing like frame interpolation or inverse telecine to recover 23.976 from 59.940 source, then all bets are off. None of those algorithms are perfect so you may see occasional glitches. Varies by manufacturer. But if the source device is not sending the right data, then you're not going to get the best results regardless of television. Garbage--in, garbage-out.