I have noticed that sometimes the music plays a tone, goes up a few semitones with it and then that last tone in the walk is treated like the fifth in the new key it modulates to. Sometimes the tone was the fifth in the old key before it started walking, so to speak, so it is kind of like the key was transposed up a few semitones. This kind of thing seem to happen in (some versions of) Bränd-Pers vals for example.
Other times the last note in the walk is the old key's leading tone, but in the new key it would be the third or something like that. This kind of thing seems to happen in Carmen by Sarasate Op. 25 for example, although it might be numbered differently because I guess it might not be the usual major key and that seems to affect what number or name you give the notes.
I am wondering if there is some established nomenclature about these kind of things. I can imagine the number of possibilities growing quite chaotically so I don't expect every possibility to have a name.