RAND impressed me by actually reading their online applications. They read my materials (papers), resume, cover letter, and recommendation letters and felt that my work was a good fit with their current need. Throughout the process, I felt that idiosyncractic fit -- the folks hiring me liked my work and found it interesting and relevant to their work and that made all the difference. If I'd been similarly skilled but hadn't done work of this type, I doubt any part of the process would've been as smooth.
First round interview was via phone with someone who found my work interesting. Partially the interview tried to sell me on the job. But in terms of hard questions, they asked me to detail my research work that I had submitted (papers). First, to describe each succintly and it's contribution and policy-relevance. Second, to describe my thought process and steps in doing the work and collaborating with partners, to ensure both that it was my own thinking/work and to show that I had been really independent in how I initiated and completed the project.
Second round of interviews was full day interviews - individual 30 min meetings with potential peers and 1 job talk. Because they are an unstructured group I found their individual interviews fairly disorganized and unpredictable. Each person interviewed you largely independently. So some people like you and want to answer your question about the job. Others are more critical and want to test your abilities to independently come up with research ideas. Others want to assess whether youd' be willing to chip in on projects outside of your area of interest but where there is funding.
RAND is a place where you need to be very independent at pursuing funding and research opportunities. The whole place is a loose network of researchers so it's all about first being independent enough to find and create projects with them, and second be able in the long run to build a niche of grant-generation for yourself and your type of projects. Methods matter but ideas and independence matter too. And the desire to spend alot of time pursuing opportunities and to branch out into whatever topic areas might have more funding at any given time.