Eb is certainly a foreigner in G major. To help the ear not to be surprised by it, you could use D# a couple of times as a chromatic passing note from D to E (D > D# > E) in preceding melody. This could be part of an innocuous harmonic progression like I-IV. You could then write a melodic fragment D -> Eb -> G, introducing C minor around G. You could tonicise it then and there, or later.

Mozart was a master of musical puns like this. See how crafty he was in the 24th Piano Concerto:

I would recommend, as a general approach, looking at Max Reger's Modulation. C min is relative to Eb Maj, a cadence from GMaj to Eb Maj could be modified to move to C min in a fairly straight forward manner.

Using the notes that make up Bo works well. Diminished harmonies are often used to move out of one diatonic place to another.

G major is the dominant chord of C harmonic minor, so I don't think you actually need to do anything special to go from one to the other. G already moves in to Cm easily. Try a progression like G, Am, D7, G, Cm, Fm, G, Cm, for example. (You could of course drop the harmonic minor after the switch, so the 2nd half of that example could as well be e.g. Cm, Fm, Gm, Cm.)

Other, shorter examples, with 3-note voicings:

• 
  

  D7, G, Cm

  
  
  

  

  
  


• 
  

  Am7, G, Cm

  
  
  

  

  
  


• 
  

  G, G7, Cm (also G, Bdim, Cm)

  
  
  

  

  
  


• 
  

  G, C, Cm (a little different)

  
  
  

  

  
  

Some of those work fine, even with just 2-note voicings.

Emphasizing the leading tone resolution (B in to C) somehow can also help make the G, Cm change sound more deliberate.

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.

EDIT

Forgot to load an image yesterday. I added a third example trying out two converging chromatic lines.

Modulations should make use of pivot chords to make the modulation smooth.

A pivot chord(s) put simply is chords that fit in both the key you are modulating towards and the key you are coming from.

so for instance, if you have a 4 bar phrase in C Major. Let say your phrase ends on an interrupted cadence. Now that six-chord of C major is an a minor chord, this is conveniently also the tonic chord of a minor. Both these chords are a suitable resolution to your cadence.

Now we are in the second chord of your second four bar phrase now you can go to the d minor chord that is again both C:ii and a:iv this is the second pivot chord you can use.

Then finally we would just have an E Major chord with a raised g# tha resolves correctly to now make it clear that we are in a minor. There is nothing that makes modulations as clear as Leading Tones resolving towards a Tonic.

Some modulations lend themselves more towards the use of pivot chords than others but the modulation to the relative minor is an especially easy one to use them with.

The progression I I7 IV iv is quite common. If you were to establish that as a pattern in G major, but gradually shift the rythmic emphasis away from the I and IV and toward the I7 and iv, then those two chords could come to be perceived as the V7 and i of C minor, without there necessarily having been any clear point where the piece switched key.

In case anyone's interested: I've solved, to my own satisfaction, the change of mode part of my question, in a rather unexpected way, by transitioning first to F sharp Aeolian. Of course, I'm still way off C minor, but modulating through minor-like modes and keys makes it easier (I hope). I think the main lesson here is that key transition doesn't necessarily involve advanced "chord magic" (although it can). Stylistically speaking, it's more about setting up proper expectations and then refuting those in a convincing way than anything else.

The repeat bars all involve two repetitions instead of the usual one, by the way. The key of G major had been very firmly established beforehand. I want to end with a modulation from C minor back to D minor (hence the key signature), followed of course by a protracted cadence in that key.

Follow-up question (to all you harpists out there): is the stuff I wrote for the harp at all playable, assuming a reasonably competent harpist? (The tempo is very slow, like 45 BPM.)