How do we keep track of the positions of stars that are light years away?

The simple answer is that, in general, the movement is not fast enough to make much difference on human timescales.

Yes, we do have to measure the proper motion of celestial objects - this is their rate of tangential motion on the plane of the sky (in units of arcseconds per year, or similar). For some (nearby) objects, this can be large enough that there are noticeable differences in position over years or decades. Fortunately, the larger the proper motion is, the easier it is to measure.

How is that done? Well, if we use very distant quasars as a "static" reference frame (the expansion of the universe moves things radially, so does not affect their position on the sky), then the motion of closer objects can be measured with respect to this reference frame.

This is essentially what the Gaia satellite has done over the whole sky. The results are presented in a big catalogue, so whenever you need to know the exact position of an object now, you can work that out from the positions and proper motions listed in the Gaia catalogue.

Your question seems to be mixing up two different problems.

Yes, stars are in motion, in extremely small ways. We used to refer to "the fixed stars", meaning they don't move while planets and comets and so on move in front of them. Certainly, we learned that stars and galaxies are actually all in motion, not fixed at all; but we can largely treat them as if they are. Their relative speeds are minuscule compared to our view of the sky.

The movement of celestial bodies only really matters in terms of how far you have to rotate your telescope to keep the object in view. If we use your pebble analogy, a pebble six inches from your face that moves a foot to the left requires a large change the angle of your view to re-center it, while a pebble a mile away that moves the same foot to the left has likely not even moved out of your telescope's view, and if it did you'd need only a tiny adjustment to re-center.

On a more astronomy-focused scale, if your telescope's field of view contains dozens of distant galaxies, one galaxy moving an entire galaxy-diameter to the side has not really changed anything; it's still well within your existing field of view. And since nothing can move faster than light, and these galaxies are tens or hundreds of thousands of light years across, the worst case -- a galaxy moving perfectly perpendicular to our view -- would take tens of thousands of years to move even one diameter to the side.

The real problem isn't the stars moving around, it's Earth. Ground-based telescopes have to rotate at the same speed as the Earth (and in the opposite direction) to track an object -- from our perspective on the surface, the stars rise and fall over the course of a night. Orbital telescopes have to do a similar adjustment as they go around the Earth, or compensate for movement around a Lagrange point.

Those compensations are well understood and calculated, but the smallest of them is vastly larger than anything the stars are going to do in the course of a night, or even a year.

The only reason that movements of pebbles can put them out of view is that there are big rocks in the foreground. This rarely happens in astronomy. However, to complement the other answers, it is sufficiently interesting that when it does happen it has its own name: occultation.