Before wire-transfer technology existed, how were funds moved from one person to another?

The earliest form of money-transfers were done by the Knights Templar, who are accredited with perhaps the earliest form of banking & money transfers.

Although the primary mission of the Order was military, relatively few members were combatants. The others acted in support positions to assist the knights and to manage the financial infrastructure. The Templar Order, though its members were sworn to individual poverty, was given control of wealth beyond direct donations. A nobleman who was interested in participating in the Crusades might place all his assets under Templar management while he was away. Accumulating wealth in this manner throughout Christendom and the Outremer, the Order in 1150 began generating letters of credit for pilgrims journeying to the Holy Land: pilgrims deposited their valuables with a local Templar preceptory before embarking, received a document indicating the value of their deposit, then used that document upon arrival in the Holy Land to retrieve their funds in an amount of treasure of equal value. This innovative arrangement was an early form of banking and may have been the first formal system to support the use of cheques; it improved the safety of pilgrims by making them less attractive targets for thieves, and also contributed to the Templar coffers.

Based on this mix of donations and business dealing, the Templars established financial networks across the whole of Christendom. They acquired large tracts of land, both in Europe and the Middle East; they bought and managed farms and vineyards; they built churches and castles; they were involved in manufacturing, import and export; they had their own fleet of ships; and at one point they even owned the entire island of Cyprus. The Order of the Knights Templar arguably qualifies as the world's first multinational corporation.

Source: Knights Templar

Earlier references to a parallel settlement network was that of Fei Ch'ien, affectionately called the Chinese Flying Money. It is estimated that Flying Money today, still represents one of the largest money laundering methodologies (more so than the term stuck with us now, Hawala).

The Fei Ch'ien was also in some aspects a legal note, which could be traded for hard currency when presented.

Almost all the settlements were based on a trusted network. Which even before the the days of the 20th Century was done in the same manner as it was 100s of years ago. To pay out, based on the limits assigned and agreed upon, and then to settle amongst the trusted parties (i.e. the early day banks) at a later date.

The seals and encoded information on these notes made the entire eco-system run, not to mention the periodic meeting of the bankers/lenders.

Physical movement of money would happen monthly which was then changed to weekly.

A great understanding of how the Automated Clearing House (ACH) system in the US works (pre wired times) can be learned from this podcast by NPR, on their Planet Money show:

Episode 489: The Invisible Plumbing Of Our Economy