they do not go to the fed, ever.
The mint issues coins to a contractor like Brinks and Lumis. The banks buy the coins from them. The bank has a policy that they will keep X number of rolls on hand, if they have X+ rolls they are returned to their contractor on the weekly exchange.
The Fed, Federal Reserve, sets policy, the discount rate, the overnight rate, and the prime. None of these have to do with coin distribution.
Ex.
I do not know the exact rates and policies any number given is for illustration
Policy- The Fed sets what amount of assets banks must keep on hand. You deposit $1000 the bank must keep $1 on hand and loan out the other $999
Discount rate- What the Fed charges the bank. The bank borrows $1000 from the fed The bank receives $990 and must pay the Fed $1000
Overnight rate_ What the banks charge each other for overnight loans to cover short terms cash shortages
Prime rate - what the best customers get (but no one gets it they all pay prime plus some percentage it may be an infinitesimal amount)
Disclaimer:
This is written from memory and I have not dealt with such things since college and that was a very long time ago so I have made errors but it should be close enough to show the Fed does not deal in coins