TVs can suffer from two motion problems: judder and blur. Look for both, ideally with 60Hz (console game) content, 50Hz (broadcast) content, and 24Hz (Blu-rays, UHD Blu-rays).
Do camera pans stutter along? Do fast-moving objects look blurred, short of detail or even leave a smeary trail behind them? Do you see momentary ‘freezes’ during action scenes? Do vertical lines in the picture suffer ‘doubling’ during camera pans?
Most TVs offer some sort of motion processing to counter blur and judder issues, so try and check these out. However, these processing systems can cause their own problems, specifically shimmering halos around moving objects, flickering over areas of really fast motion, and a tendency to smooth out judder so much that pictures – especially 24-frames-a-second movie pictures – are left looking unnaturally fluid.
Bear in mind that most TVs offer different ‘strengths’ of motion processing, so try adjusting the settings to get a more comprehensive idea of a TV’s motion performance.