Your org goes agile.
You decide to make every process explicit. My understanding is that each process is a kanban board. One organization will have many processes and thus have many boards with work flowing through.
I am trying to map teams to processes (boards) and determine which is higher in the hierarchy.
Does each team own a set of boards (e.g. processes)? This means each team is autonomous, capable of managing and burning down its own backlog.
The alternative is that one board needs serviced by members from multiple teams.
In the first case processes have to be segmented according to team boundaries. In the latter ownership of the process (board) becomes vague. When there's no clear owner responsibility is less likely to be taken.