I had lots of screening interviews both on campus and at meetings. I had on-site interviews at companies in the semiconductor, pharmaceutical, and chemical/materials industries (oh the stories--industrial hiring is no more efficient than academia in my experience). I still had nothing lined up.
I was invited to interview for the postdoc position I eventually took at National Lab after my future postdoc advisor saw my CV on one of the science sites (that is the dumb luck part). Postdoc Advisor was looking for someone who could do the type of measurements I specialized in on a system completely different from my PhD field. National Lab needed my skills, I was sick of my sub-field and wanted to change research directions, so it was a match.
National Lab flew me out for a 1 day interview. This was similar to how I've seen academic postdoc interviews described--I gave a seminar on my PhD research, interviewed with the PIs of my potential project, and met other postdocs/staff working at the lab. As I later found out a a postdoc/staff member, the other postdocs/staff had veto power, but only for a specific reason. The decision was made almost exclusively by the PIs. I had lunch at the lab, but was on my own after 5 pm. It certainly was less tiring than either the TT or industrial on-sites I went to, and I got to see a lot more of the area, since I had free time the afternoon/evening before and evening after my interview.
A week later, I was offered the job. My offer was contingent on me applying for and receiving a National Research Council Research Associateship, which I did. If you are interested in getting one, I highly, highly suggest you find a mentor and work backwards (like what I did). I know that there are lots of people who cold apply to these programs (after all, the pay is awesome, and the science is hot), but every single postdoc at National Lab had one of these (or something else similar), and every single postdoc I have ever met there was recruited first, and applied second. I have never met a postdoc who applied for a research associateship without getting a mentor first, even at other national labs. I think it is a waste of time to apply first. I can't emphasize that enough (because the application is kind of long and annoying).
After I took the job, National Lab paid for my move. I moved and started working. I worked 10-12 hours a day, but only 5 days a week most weeks, and learned how to work much more efficiently so I could keep my weekends free. About 15-25% of the postdocs at National Lab are offered staff positions at the end (mostly to replace retirements--the average age of employees at National Lab is high). They are really into "try before you buy" because firing someone is very, very hard (this seems to be true everywhere--I know lots of industrial scientists who complain about deadwood). More established people who are beyond the postdoc level are brought in as contractors to make sure it will work out before being offered a staff position, at least at National Lab. I've heard this is pretty common at most national labs, but I only know my lab really well.