When a package arrives a USPS or other sorting facility the package usually goes through the sorting facility within a few hours or within 10 hours of it arriving there.
A truck driver has the packages or another delivery driver has the packages and drops them off at the sorting facility.
In the case of USPS a large truck usually backs up to the dock of the USPS sorting facility where workers then use forklifts to unload large bins that have your package in it.
Then they take the bins with the packages to a sorting machine where the packages get sorted and then put on the correct truck for delivery to another sorting facility or to your local post office.
When I check tracking information my packages are usually sorted and loaded onto another truck and on the way again within 10 hours or 24 hours at most.
So the packages should not stay too long at the sorting facility.