I am not very clear about your question. I wrote a paper on the relationship between spatial scale and spatial institutions of government, which involves the issue of municipal autonomy and specialization (in the sense of single-purpose government and general government). The geographic context is largely set in China vs. USA. Attached is the PDF file of the paper. Hope it might be useful to you.
In my view, local or municipal autonomy in policy making is determined by at least three factors.
first, and foremost, there are legal regulations that determine the degree of autonomy of municipalities and local administrations. Municipalities usually are at the bottom of multilevel governance systems in both federal and unitarian states. So there are policy areas where no autonomous decisions can be made and the municipalities have to implement policies imposed from above, but there also might be policy areas in which local policy making is possible as long as it complies with general rules and procedures
second, local budgets might also determine the options for municipalities. Especially independent budgets (e.g. taxes raised by the municipalities) might facilitate local policy making, whereas dependence on federal funding might diminish the opportunities (especially in non-democratic states where power often is excerted through budgeting).
third, the inclusion of citizens into policy making can effects. Participation might enhance acceptance of decisions as well as it might broaden knowledge.
But there might also be other factors influencing local autonomy...
So much depends on the context, and often whether a given issue is perceived as political or technical in nature. And also whether the autonomy is staff relative to elected officials or cities relative to the senior government that crafted the legislation enabling the city's existence and /or distinguishing the city's power from that of the state, provincial, or national government's own power. For example, in most Canadian provinces and American states, city procedures for consulting with the public have to adhere to certain requirements set by the provincial or state government. But as long as cities meet these (as sort of a minimum threshold, eg minimum number of consultation and notification opportunities, etc.) they can add their own techniques as well.