Your passes
Passes lists your passes with the balance left, the validity period and a line saying "as of …".
That line is part of the balance. It says when your device last saw anything new. If you have not been in a studio for three days, the number is three days old — and the app claims nothing else.

Where the number comes from
It is not stored, it is calculated from the individual entries: the purchase, then each redemption. Every entry is signed by the device that wrote it. That is why your phone and the desk at the second location arrive at the same number independently.
You can read your passes but not change them — not even your own balance. Whoever took the payment keeps the books.
You carry the news
Your phone carries your pass from location to location. Check in at the old town and walk over to West, and that desk knows about it as soon as you come in — because you brought it, not because there is a server somewhere.

Backup
Export my passes downloads a file with your entries, signatures included. That is your receipt if something ever comes apart. It is not required — your studio holds the same entries — but it costs nothing.
A contradiction on a pass
If a pass shows a contradiction, two counters redeemed the same class without knowing about each other. Only one unit is deducted regardless. It is shown so that somebody can sort it out.