... correct, because:
(a) all the melody notes within each four-bar sequence are in those respective keys,
(b) the first chord in each four-bar sequence is one of the subdominant chords (either IVma7 or iim7) in the respective key (IVma7 of Db in the first four bars, ii7 of E in the second four bars) (which is why substituting an Ebm9 for Gbma7 works and sounds great in the first bar of the bridge; it's ii7 in Db), and
© the second chord in each four-bar sequence is the bVII7 relative to the respective key, functioning *as if* it was going to be a bVII7 backdoor dominant approach to the tonic, but which turns out instead to pivot to function as V7 in the new key in first sequence or V7/ii7 of the new key in the second sequence. You can hear the bVII7 function, as it would've sounded without a pivot, by just ending each four-bar sequence with the tonic of the respective key: first four bars, play Gbma7 ... B7(Cb7) ... Dbma7, End. Second four bars, play F#m7 ... D7 ... Ema7, End.
The brilliance is in never hitting any of those tonics, and using a bVII7 -> V7 pivot chord to change keys.
Same analysis applies to the third four-bar sequence, in F: Gm7 (ii7) ... Eb7 (bVII7) ... Here, though, the bVII7 doesn't pivot to become V7 of a new key, it just sets up a iii7 - VI7(V7/ii) - ii7 - V7 in F: Am7 - D7 - Gm7 - C7.