How do modern LCD screens multiplex?
All modern high-resolution LCDs are "active matrix", which means that there's a thin-film transistor (TFT) located at each subpixel that is used to store its state for the current frame. The state of the subpixel is stored as a voltage on a capacitor, which is either a separate element or the capacitance of the liquid crystal cell itself.

This means that refreshing the display does not inherently introduce any significant flicker. However, there is a certain amount of leakage associated with the capacitor, and an LCD designer needs to make a tradeoff between minimizing this "droop" by making the liquid crystal respond more slowly, or minimizing the "smear" introduced in fast-moving images by making it respond faster.