Changing KEY

Working Example: (Changing from Key of G to D)
Bm, Em, C, Am
Bm, Em, C, A,
Bm, G, D, A

1. G, Am, Em, C ~ Have melody for this but difficult to match it to next part

G Am, Em, C
The wilting flower, slowly grows again
The cloudy skies, are looking brighter now
Stormy seas, are calming to a breeze

Sounds Like it works (Changing from Key of C to F)

2. C, Dm, Am, F <Thought this was a change of key from 1. then realised G & C share these chords…
3. C, Dm Am, G
4. Bb, Am, Dm, C <Switches from V chord of C to I chord of F

(Switching from V chord to I chord of next key?)