Your example is not a good one. Words are often used in different forms in poetry (license). It even happens in Esperanto.... Kaj unu por la eta knabo, Cxe la strata fin' ... My gripe is not with people using "fast" as an adverb, when it is really an adjective. I am grizzling because Duolingo marked my "quickly" as being wrong, which it most definitely is NOT!
I've never seen a preposition before an adverb. I think it has to be either "Ni marŝas rapide hejmen" or "Ni marŝas rapide al nia hejmo". Or maybe "Ni marŝas rapide en nian hejmon" and "Ni marŝas al en nia hejmo" are also acceptable. But the only one I've ever seen is the first one (hejmen).