This is a common reaction when people first complete the ITIL Foundation course. They are inspired to move in the right direction. However, this direction is counter-productive to the concepts of IT service management.
While some processes, and sets of processes, may warrant a dedicated team to perform that process, processes are simply sets of activities and do not operate independently. Creating separate teams to support the processes simply moves from one silo approach to another. It is important to understand how the processes interact and how a single individual may perform the activities in multiple processes within the span of a few minutes, or even seconds.
Instead of focusing on the processes themselves, focus on the procedures that depend on the activities of the pertinent processes. These procedures can then be ingrained into the fabric of the new organization.