What does "arrange to dispatch" mean? Does it express multiple things of action?

Arrange to dispatch means to meet the requirements/prepare for transfer (of something, e.g. "5K").
The transferring itself is implied as well, but technically not part of the 'arranging'.

It can express multiple actions, since it implies that one takes care of whatever needs to be done (the requirements) for the dispatch (the transferring, sending, or mailing).
Taking the mailing of an item as an example, it needs to be properly packaged, the complete address of the receiver needs to be clearly indicated on the parcel, it needs to be passed on to a post order company, the sending needs to paid for, &c.

It will be inappropriate to write "Please dispatch..." to say the Head of Dispatch section. Instead "arrange to dispatch" possibly means "take appropriate steps to have the item dispatched..." "Arrange to dispatch" is frequently used in Indian offices. It may mean "steps to ensure dispatch..."