Considerations When Serving on Thesis Committee
Internally within the committee, asking the committee chair is the best option. The chair should be able to fill you in with the procedure and what kind of angles the questions usually take. Be aware that the chair may still insist that you should decide what kind of questions to ask. And from my experience serving in committees, they did mean it.

Externally, since your institute is in the same city with the host, why not go attend a public defense there? That way you can also get the tone right. Do know that, however, tone/attitude can still vary greatly within an institute.

As for what to ask... first, I'd say don't ask any structural questions that would destroy the student's work. Those should be cleared up before the committee signed off the student to present. Second, unless it's for clarifying some question in the results or data, use open-ended questions. From my experience, I usually ask 1-2 clarifying question (e.g. on a limitation they suggested, etc.) and 1-2 "what if's" (e.g. what would you have changed if you can redo this work?) or "how should this work be carried forward", etc. That's just my style, I like to start by making it feel like the candidate is a student answering questions to end with treating the candidate as a new expert who can make recommendations and broad takeaway.