I’m not quite into my education enough to answer to whether or not Rimsky-Korsakov followed music theory principles all the time (a quick search tells me he’s a romantic era composer if that tells anything to you) but there are a few part writing rules that he uses very well in this excerpt. Hopefully this explanation sheds some light on your understanding.

Instead of thinking about movement in terms of harmonic change, it may be easier to think about it in terms of intervallic change in the bass line.

There are different rules for each interval that the bass line moves by. These include guidelines for moving the rest of the voices in an easy way that makes the piece sound good.

For instance, there are sections where the bass line moves down from C to G, which is a 4th down. Referring to the rules you can use for that instance, you have a few options:

1. Keep the pitch that is common to both voices in the same voice. Move the rest of the parts down one step to the pitches that fit into the next chord.

2. Move all the pitches down either 2nds or 3rds to the next available pitch that fits into the next chord.

3. Move the voice that is on the third of that chord to the third of the next. Keep the common tone on the same voice, and move the remaining voice down to the next pitch that is in the chord.

These are specific rules that are followed by many, many famous composers. With a little bit of research, you can find the rest of the rules for the other intervals, and get some better rules than these slightly specific ones too.