The motivation underlying autosegmental phonology is that phonological processes frequently are not well-described by the 1-to-1 feature-to-segment relations assumed in SPE phonology, especially in the domain of tone. OT was motivated by a desire to reduce phonological mappings to just the notion of constraint, without any (language specific) component of "do this" (only "don't do that"). Neither theory has anything to do with the notion of deriving phones from phonemes, and neither theory assumes that "phone" or "phoneme" is a valid concept. Both theories are logically consistent with the concept of "phonetic implementation" which may provide pronunciation details such as nasalization in English, devoicing of sonorants after aspirated stops and fricatives, and so on. However, both OT and ASP are grammatical theories which deal in categorial processes, and it is generally recognized that most allophonic variation has to be dealt with phonetically and quantitatively. (ASP and OT phonetic models do, of course, exist, just as there is such a thing as OT syntax though not autosegmental syntax)
There are no rules in OT, so OT rules aren't based in autosegmental phonology. OT practicioners have largely (but not entirely) abandoned autosegmental phonology or any other theory of representation, partly because questions of representation are orthogonal, and partly because it was held that the motivation for ASP was how it allows a better account of phonological rules (correct, I would say), and therefore if the theory presupposes rules, it must be wrong. At this point, I don't think many OT practicioners care about autosegmental phonology, one way or the other.