Short answer
Give people from different departments one shared way to work a problem, one owner and the authority to decide. Cross-department problems stall when each team defines the problem differently, nobody owns the answer and every decision has to go up the line. Agree on a simple shared method, put the people closest to the work in the room and let them decide within clear limits.
The longer answer
Most companies have good problem solvers inside each department. What they lack is a common way to solve problems across them. So meetings turn into each side defending its version of what happened.
A shared method doesn’t need to be complicated. Agree on what the problem is, in one sentence. Look at the facts together, where the work happens. Agree on the cause before anyone proposes a fix. Then name one owner and a date.
The managers who run this well become the people who teach it to others.