To send with purpose and address — a message or a doing released toward its handler.
Italian dispacciare or Spanish despachar (to expedite, hasten), a shared Romance formation whose deeper root is disputed, generally read as dis- plus a form of Latin pactus (fastened), the sense being to free something to go on its way.
A sending with purpose and address, whether a message or a piece of work released toward whatever will handle it. What is dispatched leaves the sender's hand the moment it is sent.