OH Boy! This is a bit of a hot button for me.
Years ago I did a direct-transfer of an employer sponsored HSA to a 3rd party. It took months.
Never again did I do a direct transfer. I always used 60-day rollover/ aka indirect transfers where you just withdrawal the funds from the original trustee and deposit the funds with the new trustee -- being sure to have the new trustee code it as a 60-day / indirect rollover.
Note that some retirement account transfers (ie inherited IRAs) may not allow for direct transfers so proceed with caution if you have not done indirect / 60-day transfer/rollovers in the past.
Also be warned that, in general, there is a limit of home many indirect/60-day rollovers that you can do per year -- at least for IRAs.
-gauss