"To take over the responsibility" means to accept or assume the duty or obligation to complete a task or to take care of something that needs to be done. Full definition
That approach only works if your systems ensure someone else takes over the responsibility for forging great relationships with existing customers. (inc.com)
After a borrower passes, heirs take over the responsibility of repaying the remaining reverse mortgage loan balance. (americanadvisorsgroup.com)
Before I left she brought me a box containing the communion utensils and suggested that someone else take over this responsibility. (religion-online.org)