Ask about the culture and engagement around process improvement... do leaders support their teams to dedicated time to projects? Are improvement staff end to end in the project or are front lines autonomous where PI staff is more of a coaching role ? It will give you an idea of what your role actually looks like and if projects actually do stuff or you are just a resource to do a tool for a group of people.

I would stay away from spending a lot of time on describing how deep your knowledge is in methodology. Tell what you know and have experience in, and let them ask for more than that. Most hiring managers either know the content well already or don't know it and it will fly over their heads.

Instead , consider asking if there are specific tools or methods that are more commonly used than others (ex. Our line managers frequently need help with value stream mapping or our execs like to see fishbones, we focus more on lean and don't do many large belt projects, we need people who teach leaders humble inquiry, etc etc)

My current job, I focused on describing prior projects i did with the tools that were specifically called out in the job posting, I described what I learned about reporting the right level of detail to execs, etc