Well, no, you're not quite getting it. Moving North doesn't make skin whiter (if that were true, there wouldn't be any black people in the US). Rather, mutations in genes, favored by natural selection, make skin whiter.
This study indicates that the mutations of interest occurred later than earlier hypotheses had suggested, but long after the first human beings migrated to Europe.
Compare the skin tones of an Afro-American to an African. People whose ancestry dates back to antebellum America are distinctly paler than recent African immigrants.