I need a little more explanation of the nuance of the situation to decide on which word to recommend.
The first word that came to mind is entrust. Oxford's Living Dictionary defines it as
1 Assign the responsibility for doing something to (someone)
1.1 Put (something) into someone's care or protection
I hesitate to recommend it, though, because entrust is not generally used in a business context.
Delegate is more commonly used in a business setting, but only for tasks. If I am a manager and need to get something done, I can do it myself or I can delegate it to someone I am managing. It means pretty much the same thing as entrust but it carries the extra connotation that (a) it is a task or obligation that is being given to someone else and (b) the task is something assigned to the first person by someone else. It is also only used for tasks and responsibilities, not physical items. I can entrust someone with my computer, but I cannot delegate my computer to them.
Assign is also similar, but is usually for something new, and can also be used for physical objects. However, assign connotes giving someone ownership (to some degree) of the thing.
A customer places an order on an e-commerce website. The software assigns me the task of filling the order. I delegate the task to a trainee I am supervising.
In many cases you could use transfer, which means to move from one person or place to another. The distinction here is that there is no implied obligation on the recipient. If I assign or delegate something to someone, I expect them to take care of it and/or get it done. If I transfer something, I'm giving it to them for them to do what they want with it.