@MMazzon, it is apparent looking inside my piano that there are many different gauges of string. The lowest strings have visibly different thicknesses. Each set of strings (for each key) has a different length, like a harp. I would not call that a gradual change, but graduated.

@Tim, the highest strings do not even have dampers since they vibrate for such a short time, so the sustain pedal does nothing to them.

This is what I have seen in the wild, and what I have used myself in writing my scores when I used an equivalence like this to change keys. It helps the singer's ear to see what the note is in the new key for reference.

I disagree with your use of "segment" and "sequence." A segment is more of a reference to a group of figures that make up a piece of a measure or measures, not an entire phrase. "Sequence" has a very specific meaning, which is a segment of music (using the definition I just gave) that is immediately played again, transposed up or down by a certain interval. Sequences are often, though not always, found in groups of three. A phrase can contain sequences, but a sequence by itself does not necessarily make a phrase.

Can you put up a pic of the music you are playing and tell what fingering you are using to cover those notes?

@Tim, I figured I would tag it onto your good answer, even though it didn't directly relate to the OP's question.