The arpeggio continuously plays in E minor while the chords in the progression change.
Should I adapt the arpeggio on the progression?
Music is art. You can do whatever you want.
But, it seems clear to me you are trying understand some basic harmony concepts like combining chord tones and non-chord tone to follow a harmonic template (chord progression.) You're trying to learn about musical conventions.
Two concepts that you should look into are pedal points and ostinato.
A pedal point is when a single tone is held for a long time while harmony in other voices continues to change, and importantly, some of the continuing harmony deliberately clashes with the pedal tone! Usually the changing harmonies return to match the pedal tone, and the commonest pedal tones in tonal music are the dominant and tonic. From a certain "high level" analysis the harmony of a pedal is just an elaboration of a dominant or tonic chord where the whole passage can be analyzed as either
The connection between your idea to a pedal is that you have one thing - the
E minor chord - being held while the other parts move against it. You have a sort of "pedal chord." If your chord progression came back to
E minor, like
Em-C-B-G-Em or something similar, it would be following the concept of a pedal but with more dissonance from multiple clashing tones.
You can generalize this idea even further. You mentioned that your
E minor arpeggio uses passing tones. The super high level concept is the ebb and flow of consonance to dissonance to consonance, stability to instability to stability. Passing tones, pedal points, tonic to dominant to tonic, etc. exhibit this idea. It's a basic fundamental of tonal harmony.
Ostinato is when a fairly short melody idea is repeated over and over. If your
E minor arpeggio not only holds the chord, but also repeats a short figure, it would be an ostinato.