"Delegate responsibilities" means to assign tasks or duties to someone else, giving them the authority and accountability to complete them. Full definition
The state delegates the responsibility to local - what they call local service districts - basically, cities. (gothamgazette.com)
I agree that authors need to get expert help, but it's worth noting that they can't delegate responsibility for the contract to their agent. (accrispin.blogspot.ru)
While the CT State Department of Education and State Board of Education delegated the responsibility of implementing NCLB sanctions to local districts for schools under local control, they apparently haven't assumed that responsibility for schools under their own supervision in recent years. (jonathanpelto.com)