Different voicings for different chords
A different set of voicings, derived from the appropriate bebop scale, is used for each chord. Over the I chord, the voicings come from the major bebop scale and alternate between the major I6 and vii°7 chords. This is because the stable scale degrees (1, 3, 5, and 6) tend to sit on the downbeats, so we'll hear the sound of the I chord on the downbeats and a passing chord, vii°7, on the upbeats. This principle of alternating between the main chord and passing chords is central to Barry Harris's harmonic concept, and also used widely in big band arranging.
Over ii and V, we generally use the mixolydian bebop scale. In C, that's
G A B C D E F F#, adding a major 7. The two chord shapes outlined are G7 and F♯ø7, and a well-constructed melody usually again has G, B, D, and F on downbeats, so the F♯ø7 acts as a passing chord. (Occasionally, one note can be swapped around in order to create smooth voice leading; changing the E here to a E♭ to form F♯°7 is a natural choice.)
If the ii chord is held for a long time, you can also form a "dorian bebop scale" by adding a ♭6 or major 7. The specific chord voicings chosen will depend on the context, with the goal of having a ii7 sound on the downbeats and a passing chord on the upbeats.
Relationship between block voicings and functional harmony
EDIT: While writing this I came to the conclusion that the biggest challenge for me is to make coherence between the Barry Harris voicings and functional harmony. Harmonizing a Major scale in triads is what gives our characteristic chords (our ii-dorian, iii-phrygian, IV-lydian, V-Mixolydian, etc). This of course vanished when we re-harmonize.
I would say that when you use Barry Harris voicings (also called "block voicings"), you are essentially tonicizing whichever chord you are on at the moment. In the G7 example above, the passing chord is F♯ø7, which can be analyzed in C major as a secondary dominant: viiø7/V, or simply V9/V if we view F♯ø7 as a D9 with its root omitted.
When the harmony gets further away from the tonic, a good approach to writing block voicings is to try to pick a secondary dominant–type chord to alternate with the target chord. We often teach beginners to look at the corresponding bebop scales instead, but I think that secondary dominants are generally what's going on "under the hood."
For example, in C major you might have an E7 chord (as in "On the Sunny Side of the Street"). Naively, you might choose the E7 mixolydian bebop scale to harmonize this, alternating between E7 and D♯ø7 voicings. But a more perceptive arranger recognizes the E7 as V/vi in C major, and thus will choose a harmonization that sits better with local A minor context. Drawing on the A harmonic minor scale, I would modify the E7 mixolydian bebop scale's F♯ to an F and C♯ to C (implying E7♭9♭13). Then the two block chords are E7 and F7, where the latter could be analyzed as subV/V in A minor.
Here, I embellished the melody of "On the Sunny Side of the Street" and harmonized it using (mostly) block chords. Blue denotes regular block chord voicings, olive is drop-2 block chords, and red is for "something else."
Notice that in the second (full) bar, I have used the E7-F7 chord pair as described above for the first three beats. But, on beat 4, we have the note C on a downbeat, which won't sound nice with the F7. Instead, I "pretended" the melody note was a B and wrote the corresponding drop-2 E7 voicing that would've gone with that; now, altering the B to C creates a E7♭13 sound. This is the sort of subjective choice you get to make in edge cases like this.
Likewise, on beat 4+, we have the note C♯, which belongs neither to the current E7 chord nor the F major chord that follows. Again subjectively, I decided to just take the F6 voicing from beat 1 of the next measure and move it down a half step to create uniform chromatic motion. (This is called "planing," and it's perfectly acceptable in jazz even if it creates parallel fifths. We're done with freshman music theory now!)
In the third measure, I harmonized the Fmaj7 as an F6; it's perfectly fine to swap the note D for an E (restoring Fmaj7) in these voicings unless F is in the melody (which creates a dissonance).
As for the red chords, in the first one I "pretended" the melody had a D and wrote the corresponding closed-position voicing for Bø7. On beat 2, I used a nice drop-2 voicing for F13, the secondary dominant of E7.
I hope this example makes it clear that in practice, there are often a few different block voicings options available a given note, and the choice comes down to habit and personal taste. And there's no need to use block voicings exclusively—you can mix in other kinds of block voicing to punctuate certain notes and keep the lower voices in a nice range.