What if my former colleagues are in a completely different department or function from the roles I'm targeting?

Still reach out. The goal is not for them to evaluate your fit for the role — it's just to connect you to a recruiter. Your supply chain contact at Google doesn't need to know anything about product management to forward your name to someone in talent acquisition. All you need from them is a warm handoff to the right person. A recruiter connection through a mutual colleague is vastly more effective than a cold LinkedIn request, even if the person making the introduction has zero context on the role itself.