What you are experiencing right now is completely normal, even though it does not feel that way at midnight when you cannot find any information.
Transfers in the California system are handled quietly and deliberately. The CDCR does not update inmate locators in real time, and the VSA tends to lag as well. The institutional priority during a transfer is getting the inmate safely to the new facility, not updating databases. That administrative catch-up happens after the fact.
The inmate cannot make calls during the transfer process either. Until he is processed into the new facility, checked in, assigned housing, and cleared by staff, phone access is not available. That blackout period is standard and is not a sign that anything has gone wrong.
From the time of transfer, expect the online systems to update somewhere within a week to ten days. It can happen faster, sometimes within a few days once he is fully processed in, but do not count on it being immediate. The middle of the night on transfer day is the last time you should expect to see updated information anywhere.
The best thing you can do right now is sit tight. Once he is settled at the new facility he will be able to call, and that call will likely come before the website catches up anyway. If more than ten days pass with no update and no contact, then it is worth calling CDCR directly to ask for a status confirmation.