Not much of the controlling of a cost budget is about cost. Here I am referring to a department's cost and not product or investment cost. A department is basically a group of people who are put together to achieve something, be it a set of projects or repetitive activities.
The first thing you can think of controlling here is the cost. So let's look at what that cost is. A department's cost is salary, travelling, consultancy, some activity-related expenses and perhaps the tools people need to do their work. The team is allocated a budget which is an amount of money for doing what they planned to do.
When having a business review, the usual thing is to go through the cost variances between what has been planned for and actual spending and discuss plusses and minuses. Easy way to go and mostly a waste of time.
What really matters here are the activities: is the team delivering on what was promised? To be able to answer that question, there needs to be a plan and a clear view on deliverables. The team lead needs to know which direction things are going and what should be done by when.
So, when having the next business review with a department, ask the project leads to prepare a brief status on their main activities and start the discussion with that. It won't be easy if the team is not used to it, but it is the way to go if you want to do good work. At the end you can spend 5 minutes to see if the salaries are still in check.