In order for the reward of charity on behalf of the deceased to reach him, is it essential for it to be given from the wealth of the deceased?

There is nothing wrong with that, and the reward will reach the person for whom it was intended, if that money is spent for a good purpose such as charity, upholding ties of kinship, jihaad and good works that will benefit the Muslims, such as a person giving charity on behalf of his grandparents, even if they did not leave behind any money, or on behalf of his friend. The deceased will benefit from that. And Allaah knows best.