No, the money does not follow them automatically and it will not be waiting in an account at the new facility.
Here is what actually happens. Before transport, or shortly after arrival at the new facility, the inmate will be asked to provide a mailing address where the remaining balance from their old account can be sent. That money gets issued as a check mailed to whoever they designate, typically a family member on the outside. It does not move electronically from one facility to another because county jails almost universally run on different commissary systems that have no connection to each other. There is no transfer mechanism between them.
The practical implication is that there will be a gap. Between the time he leaves the old facility and the time a new account gets funded at the new location, he will have nothing on his books. Depending on how long the check takes to arrive and get deposited, and then how long it takes to get money back onto his new account, that gap can stretch to a week or two.
If you want to make sure he is not starting from zero at the new facility, the best move is to put money on his new account as soon as you have confirmed the facility and the deposit method they use. Do not wait for the old balance to come back to you first. Get something on his books at the new location right away so he has commissary access from the start, and treat whatever comes back from the old facility as a reimbursement when it arrives.