Look at this situation.
Mike and Mary works in a same company they are salespeople and noone has higher rank than the other.
Mike has some Chinese customers who can not speak English. However, Mary does speak Chinese. So, Mike gives the Chinese customers to Mary so that Mary can work with them.
Is it correct to say "Mike changed his customers into Mary's" or "Mike delegated his customers to Mary"?
Delegation is when you give someone else a task to do for you. It would normally be a superior that delegates work to his subordinates. And we normally refer to pieces of work, tasks, jobs etc being delegated - not people, like clients.
If the arrangement is that Mike has completely let go of the Chinese-speaking customers so that Mary can handle them and perhaps get any benefit or reward that comes from that, such as commission, then she's not really handling them on Mike's behalf. He hasn't delegated - he's passed them over.
Mike passed his Chinese-speaking clients over to Mary.
The word delegate specifically concerns responsibility. As a noun a delegate is a person to whom a responsibility has been transferred. As a verb it means to transfer responsibility. So Mike delegates the handling of his non-English speaking customers to Mary.
The real question is, why would Mike send his non-English speaking customers to Mary who speaks English. Not funny, Mike! :)
Mike reassigned his customers to Mary.
is what I would say or write.
To use the word in your example, you could say something like
Mike delegated responsibility for all Chinese customers to Mary.
