Just yesterday I sent in the paperwork to roll over my HSA from one bank to another-- the punchline is, my insurance provider didn't change and my insurance login was the same (Premera ConnectYourCare), but it looked like the 15k in my HSA account inexplicably disappeared. Needless to say, this was concerning.
A few phone calls later, and I come to find out that my 15k with Bank_0 was available on the backend, but since Bank_1 was preferred by my new employer, Premera shows Bank_1 on the frontend. Keep in mind, all interactions with your HSA bank account are abstracted away from you. So you must track down your Bank_0 and Bank_1 account numbers (I used my tax paperwork) and mail in honest to god physical signed forms authorizing the rollover.
Then I come to find out that I should have done this anyway-- HSA accounts are laden with monthly account fees, paper billing fees and investment fees that my previous employer wasn't paying anymore, and these aren't covered by my current employer. HSA bank accounts are one hell of a moneymaking venture for the big banks-- 0 risk because employers send yearly contributions into the account; high likelihood that people will forget to roll it over and the account will drain via fees.
HSAs have the hugely hidden cost of managing this separate account+tax entity, and I did not know this when I signed up for one. I'm not even sure that my hours of labor on it are worth the tax savings.