For us fit interviews make sense enough that we started doing them for every candidate, instead of just for ones that passed the technical parts, and I'll try to explain ... Maybe you can see something that could be useful for you.
There's ~1000 people in our IT (and we are hiring - poke me if your interested), 50+ different nationalities, 95% of us are expats, we mostly do Perl code on BackEnd, expect everyone to have good commercial awareness (put ideas on backlog, push back on ProductOwner ...etc) and overall try to have the least bureaucracy/processes as possible.
You can write code, push and roll it out into production within 30ish minutes - with no stamp/approval required (though on call sysadmin will ask you why are you doing it at 6am on a Sunday).
IMHO quite an interesting setup for being majority part of publicly traded group (Nasdaq: PCLN).
Most often you're not interviewed by people you'll work with (interviewers are not from same team).
Fit interviewers focus/check commercial sense/awareness, motivation, flexibility regarding approaching things like tools and technology (we don't introduce new technology unless you can make a good commercial argument for it), follow up on anything raised by recruiters and technical interviewers, and also answer candidates questions.
Indeed most of reasons we would not hire someone are noticed in other parts of interview as well. Besides being the ones who answer the most questions to candidates (besides recruitment) - fit interviews tend to be helpful for cases where everyone else is on the fence.
If someone aced the tech parts - it's extremely rare fit interview is so bad we don't make an offer.
BTW - everything mentioned is my personal interpretation ...
If anyone has specific questions I'll try to answer them.