Ultimately, unless they are holding physical currency (undoubtedly, they have some), the dollars are held in an account at a correspondent bank in the US and ultimately at a federal reserve bank. If China tries to spend money they don't have, someone in the chain is going to reject the transaction.
I never thought of this and find it an interesting. Are you saying that the primary counterfeit tracking of digital dollars is a distributed ledger where each bank knows where they got money and where they sent it? Do banks share these ledgers with each other or the fed?
I think it helps to go back and think of money as a physical object. I have some cash which I deposit at a bank. The bank holds the cash in a vault and gives me a statement telling me what I have. If I want to pay someone using the same bank as me, the bank can simply debit my account and credit the payee. The cash just stays sitting in the vault.
Now, I tell the bank that I want to pay someone at another bank using a wire instruction. My bank debits my account and tells the payee's bank to credit his account. However, my bank now has too much cash in the vault and the payee's bank, not enough. To settle the transaction, my bank must physically transfer the cash to the recipient's bank. The banks have large numbers of customers and handle large numbers of transactions, so they can net it all out and only transfer the difference.
Still, the banks don't really want to be transferring large amounts of physical currency everyday, so they deposit some of their currency with the federal reserve (think of it like the bank's bank). Then, instead of transferring physical cash, they can tell the federal reserve to transfer the money from one bank's account to another while the physical cash sits in the vault at the Fed (or more likely doesn't exist at all because the Fed is the source of truth).
This can all get several levels deep. If I wire money to a supplier in China, the Chinese bank receiving the money doesn't want to transport cash across the Pacific, so almost certainly, they have an account at a "correspondent" bank in the US which, in turn, holds an account at the federal reserve. Meanwhile, in order to facilitate foreign currency exchanges, the US bank likely also has an account at a correspondent Chinese bank . . . and I can't say anything about how the Chinese banking system works, but I'm sure it is analogous.
