With vowel sounds, there aren't really common sound changes that categorically result in a specific phone. With some consonants, there certainly are - /p/ -> [f ~ ɸ] is a good example. Common processes like lenition and assimilation are common causes of consonant changes. However, vowels exist on a more fluid spectrum, and without more context, there's no reason to say that any vowel sound more characteristically results from one change - such as raising, lowering, fronting, backing, rounding, unrounding, tensing, laxing, or something else - than another. Thus, an [ɞ] could result from essentially an infinite number of specific changes. Obviously, phonemes realized near to [ɞ] are more likely to become it, so the backing of an /œ/ phoneme, the fronting of /ɔ/, the rounding of /ɜ/, the centralization/laxing of /ɒ/, and the lowering of /ə̹/ are more probable than more extreme changes. Since phonemes realized around [ɔ] are probably more common cross-linguistically than the other options, I think /ɔ/ -> [ɞ] is a good choice if you want the sound change to be in all positions. Something similar has happened in French, where (I believe), /ɔ/ is often quite centralized.
Sound changes can be conditioned by phonological environment, encouraged by the structure of a sound system, or essentially random (i.e. the phonological "reason" for the change is impossible to determine, and sociological factors are probably more important). The "random" option cannot really yield any good answers to this question.
Changes conditioned by phonological environment mean the shift in one sound in a certain direction based on its position in a larger phonological unit (like a syllable) or based on its proximity to other sounds. Eventually, the change might generalize to other environments. Common environment changes affecting vowels include nasal raising, velar raising, and velar backing. Any of these could result in an /ɞ/ phoneme, and the specific starting sound would be up to you. One example could be /ɛ/ -> [ɞ] / _C [+ velar] [+ round].
Vowel changes conditioned by the structure of a language's sound system are famous since they can contribute to chain shifts. Basically, there is a clear psychological pressure for vowel systems to stay spaced out. This reduces the probability of speech errors such as phonemes being confused. If one area of the vowel space is crowded or one is empty, a vowel is likely to move to the empty space to stay spaced out from its friends. In your specific case, you can imagine a vowel system with /a/, /ɑ/, /ʌ/, /ɤ/, and /ɔ/. These vowels are cramped in the periphery of the lower vowel space. One can imagine /ʌ/ being rebellious in this situation and shifting towards [ɞ] to space out.