This is a subject that is near and dear to my heart. I worked in the finance department at the circuit assemblies for about 5 years. Man! did I learn some stuff there. Here are a few things in no particular order.
The WTS does not support the circuit assemblies at all - zip, nada, zilch. In fact the circuit assemblies are a good money maker for the WTS. Our little circuit sent $2000 to the WTS each assembly. $1000 each to the worldwide work and the kingdom hall building fund.
They make the announcement at noon on Sunday about how far behind they are. They usually are pretty far behind. The Sunday afternoon contributions are always the largest by far. Only once did we end up short.
Each circuit's assemblies have to be accounted for separately. IOW if you run over on one, you can't carry the surplus over to the next one. The one time we came up short, we were told that we could not just take the money out of the circuit's bank account to cover the shortfall, but that the money had to come from the congregations. So a letter went out to the congregations telling them what their part of the shortfall was. All the while the circuit had money in the bank and several thousand dollar "on deposit with the Society" - meaning we were giving the WTS an interest free loan.
Circuit overseers are reimbursed for their expenses by the congregations. If a congregation declines to reimburse him, he must report it to Brooklyn.
All of this is the way it was when I stepped down in 1995 here in the U.S. Your mileage may vary.