When I need an example of local authority, I always refer to the Catholic idea of subsidiarity, an organizing principle that states that matters ought to be handled by the smallest, lowest or least centralized competent authority. Political decisions should be taken at a local level if possible, rather than by a central authority.

However, we've seen how well that has handled the improper behavior of priests.

What tends to happen in the US is that issues tend to get ignored at the local level, and they basically go up the chain of authority until it falls in the hands of the federal government, not because they want to handle the issue, and not because they're the most capable, but because they can't pass the buck any further. And when the issue isn't handled, it's easier to blame the federal government than to identify the inability/unwillingness of local authorities to do it themselves.