It's not "walking down". It's "down a road", like "on a road". A man is walking down a road.
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In BrE, it's quite usual to say 'walk down the road' where nothing hill-ish is involved, i.e. 'toward a point away from the speaker or the speaker's point of reference'. 'Down' here is an adverb, meaning 'along' (thus 'walk along the road').
BrE says 'walk up the path' mostly in contexts where a building is at the other end of the path. Thus 'to walk up to th