There is such a thing as "tone shift", which is especially widespread in Bantu languages (see Kisseberth & Odden (2003) in The Bantu languages). Shifting can be by a vowel (mora), a syllable, or multiple syllables. Shifting by a vowel vs. a syllable are the same thing, except when you are dealing with long vowels / diphthongs, and syllable-internal shifts from one mora to the next are not common, but they exist. However, as K&O point out, "shift" can arguably always be reduced to assimilation (spread) plus disassociation, and there are numerous cases where spreading is more general and disassociation is more restricted, giving a mixed case where you have both spread and shift.
Please note that a "shift" in tone is a movement from one underlying position to a different surface position.
Your question / example seems to be more about contouring patterns within a syllable. There are two sub-parts to the question. The first is, how many tonal targets can there be in a syllable? That is kind of a highly language-specific question, but there are languages known to have three tones on a single short vowel. There has been a bit of distraction in the literature over the difference between length and duration, because it takes longer to produce a HLH contour on a vowel that to produce a simple H, so you have to apply phonological criteria to decide if a vowel is long, versus has "greater duration" (than...). The absolute max in the attested contours as far as I know is a four-tone complex in claimed to be Lomongo, but I won't vouch for that, but three-element contours are common enough.
The second part is, to what extend can the movement-points in a multi-toned syllable be contrastive. As far as we know, there is only one kind of short rise or short fall, assuming a language with just H and L tones (with 3 levels, you can have many more kinds of rise and fall). You also do not find multiple types of rise and fall on long syllables, even though [v́v̂] vs [v́v̀] would be perfectly sensible from a theoretical perspective. There may be functional and historical reasons why such a contrast doesn't exist. (There is a controversial claim about Shilluk, which would have a separate paper to review). One formal explanation for the syllable-internal restriction is that tones are properties of syllables, not individual vowels.