If a high quality baker in France sells a white country loaf and a brown country loaf, what differently will have been done to the white one?

The brown bread or pain rustique, is made from wholewheat and is untouched, bread here in France is NEVER dyed.

The color comes from what grains are used, and possibly if molasses or honey has been used. I really doubt there will be any dye involved.

White bread flour is bleached, not dyed.

A brown loaf would be made with wholemeal flour that is not bleached. 

Uh, no. The flour is bleached and ground extremely fine at the mill. FYI bread flour has much more gluten in it than whole wheat flour, so extra gluten and other additives need to be added to whole wheat bread make it rise higher to be as fine or light. (That's why they bleach the flour in the first place, the texture is better.)

A "brown" country loaf would probably be made with a Whole Wheat flour while the white would have been made from a wheat flour that has had the bran removed. 

It is also possible that the brown bread may also have a rye or other dark flour used. 

And if it was, as you call it dyed, the product used most to darken a loaf of bread is cocoa powder.  This is done a lot with pumpernickel and sometimes with the dark rye breads

No. It means a different type of flour was used.