A modulation from G major to C minor would be a modulation to the subdominant.
The most straight forward way to to the subdominant is the lower the leading tone and apply it to the tonic chord. In G major, change the F sharp to F natural and add it to the tonic G chord. So the tonic I chord in G major will become the dominant V7 in C (major or minor.)
The interesting part is your concern about the tone E flat and the possibility it could be jarring.
The vii°7 in G major is F#,A,C,Eb so you could use that to move into G and introduce the E flat. Something like G: vii°7 I Cm: V4/2 i6/3.
Another possibility is to go to the major IV first then just lower the third to make it minor iv. If that is combined with the G7 you will get a nice chromatic descent F,E,Eb plus both work with the notion of altering tones by lowering them.