After some more poking around I found something very interesting.
The Am G C sequence can actually be transposed into 2 other root keys (I hadn’t really tried them all on the last post), A# and B - or the two notes on the list directly below A (physically not musically below). Then if you change the order of the chords to be G C Am, you’ll find that you can now change the root key to any of the 4 notes on the list below G (e.g. B - G#). Now change the order to C Am G and you can change the root to any note since they are all below C in the list.
So you can only transpose to root keys beneath the root key of the first chord selected.