will it switch like this (C Ionian -> C Mixolydian) or like this (C Ionian -> G Mixolydian)?
Both; depends on what you want to do and where you want to go from the mode you are on. Let's look closely at these two examples.
We are in C Ionian mode (C major); if you go to C Mixolydian, you'll find yourself in the V mode of the F Ionian mode.
We are in C Ionian mode; if you go to G Mixolydian, you find yourself in the V mode of the C Ionian mode. If you want to stay in the C major scale, you can use G mixolydian. If you want to go to F major, you can use C Mixolydian. You can also use other Mixolydian chords to lead to other scales.
So, both of these examples are correct. In order to see which one to use in your song, just see where you want the harmony to lead to.
In this progression:
G Bm Em Dm G7 C Bm A D7
It uses chords from the G major scale:
G Bm Em all belong to the G major scale as I,iii,vi.
Dm G7 C belong to the C major scale as ii,V,I; C is the IV of G, so this is a ii V which leads to a chord in the scale we were in. This is really common in Jazz songs.
Bm A D7 belong to the G major scale again, as iii,V/V, V.
V/V means that A is the V of D, which is the V of G; so V of V or V/V.
The progression ends on D7 which is the V of G and leads back to the starting chord, which is G.