> It looks like white wins for n=6 and n=8 with perfect play, otherwise it is a draw.
For n=6 it's "1. N4 mate," right?
For n=7, "1. R5" is a stalemate; "1. R4 N4 2. N4 r4#" is a loss; "1. N4 n4 2. R2 r6 3. R3 r5 ..." is a draw by repetition; so yeah, it's a draw with best play.
I checked n=6...20. It looks like white wins for n=6 and n=8 with perfect play, otherwise it is a draw.
With random play, black seems to have the edge regardless of board size. About 2/3 of the games end in a draw, but black wins 20% and white 13%.