Will my BAH stop if I'm staying at Navy Gateway Inns & Suites (NGIS) while TDY en route to my new permanent duty station?

There are a couple of different situations that could apply, so I'll speak generally.

If you don't have dependents (if you do, I'll go down that path), it's pretty straightforward. Your BAH at your old PDS continues until you sign in at your new PDS (if your old location didn't have a BAH rate, you would get BAH RC/T (the 'T' stands for 'transit')).

This all assumes you don't have government quarters at your new PDS or some other 'out of the norm' situation applies - see exceptions in the FMR*, section 10.1.1

BAH is based on your PDS. If your TDY location were your new PDS (it isn't if you're saying it's 'TDY enroute'), you would be paid Partial BAH if you were without dependents and assigned government quarters (i.e., staying in the the barracks, an assigned VEQ room, etc). Lodging while you're TDY is handled under per diem rules, not BAH (again, this assumes you are not being assigned Government quarters at your new PDS).

To lay it out plainly:
● BAH at your old location until you sign out.
● BAH for your old location continues while TDY.
● Per diem while you are traveling at your TDY location.
● BAH at your new location when you sign in (old BAH location rate stops the day before).
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* https://militarypay.defense.gov/pay/allowances/bah_types.aspx
* DoD 7000.14-R Financial Management Regulation Volume 7A, Chapter 26 - https://comptroller.defense.gov/Portals/45/documents/fmr/current/07a/07a_26.pdf