I would say from '86 to '87 is where you really see the shift starting. In '86, the most popular rappers were Run-D.M.C. and the Beastie Boys. In '87, Rakim, Public Enemy, EPMD, BDP, and Kane all dropped their debut albums or singles. Public Enemy was supposed to drop their first album in '86, but it was delayed for months and by the time it came out, it was already dated. They heard "I Know You Got Soul" at a club, realized that the sound of hip hop was changing rapidly and made "Rebel Without a Pause" in response.
'88 was the continuation of the shift that started a year ago, but that year was when you saw more competition and more acts drop their debut albums. Sample-based production had already been around, but it became widespread in '87 when rappers started sampling funk and soul music.