The Sun does indeed drift across the sky throughout the year, not only rising higher in the summer and lower in the winter, but also varying along an east-west axis. This can be shown by observing the Sun at the same time each day throughout the year, and seeing that it changes position. This shape is called an analemma, and is a result of the earth's axial tilt and orbital eccentricity around the Sun.
Here's a diagram showing the position of the Sun at noon throughout the year, as observed from the Royal Observatory in Greenwich. Most people are aware of the change in altitude between summer and winter, but the fact that the angle of the sun varies by about 10 degrees side to side-to-side may be less well-known.
Now, this figure shows the position of the sun at a fixed time of day over the course of the year. But the question relates to the position of the sun at sunrise, which is clearly not fixed throughout the year. To answer the original question of if the sunrise "moves", you can imagine drawing an analemma for any time of day - let's pick a time that's before sunrise for part of the year, and after sunrise for another part of the year (let's say 6am). In this case, the whole curve shifts downward, and the bottom part of the analemma drops below the horizon, showing that indeed, for some parts of the year, the sun will be visible at 6am, but at other parts of the year, the sun will not be visible at 6am. Furthermore, we see the curve drop below the horizon at two different spots - this shows that there are two dates at which sunrise is at (approximately) 6am, and that the sun will rise at different azimuth on those days.
It's a little more complicated if we want to draw the "sunrise analemmas" over time, since we'll have a figure-8 shape that drifts upward as sunrise becomes earlier, following the position of the sun as it moves across the sky. It's not as easy to visualize, but that series of analemmas will drop below the horizon at different points, showing that the sunrise does move throughout the year. This shape, which is traced out by the intersection of a series of analemmas with the horizon, is not an analemma itself. The example of two dates with the same sunrise time is much easier to visualize with a single analemma.