I've got a good handle on how this developed in the southeastern U.S. with some of the tribes in that arena: The demand for British goods (particularly British-made arms) above all others was quite high among many native tribes, and this demand stretched far back into the previous century when Great Britain was nurturing the colonies on the mainland. Many recorded instances (see below) exist where chiefs specifically requested British weapons, citing their superior quality as compared to those produced elsewhere. In fact, the largest Indian trading company in existence at the time was a British firm called Forbes & Co. (formerly Panton, Leslie, & Co. until 1804) which operated out of Spanish Florida under permission of the Spanish Crown. All of its founding members were staunch Loyalists during the Revolution, and an extremely large number of traders with whom they closely worked had actually become intermarried with native women during their tenure on the Indian frontier. This is a key factor in the recruitment during the War of 1812, I would argue.

Among southeastern tribes, being matrilineal, marrying a native woman and producing children with her gave those children quite a unique new identity. They were still of full status within their mother's clan, but also had the benefit of being able to learn English (and many did) through their father, giving them an extreme advantage in terms of political mobility. The ability to negotiate in English provides avenues to those trade goods I mentioned earlier, and controlling the flow of trade goods/weapons had always been a powerful position in tribal communities. When individuals such as George Woodbine and Edward Nicholls of the British royal navy landed on the gulf coast to recruit Creek, Seminole, Choctaw, and Chickasaw allies, they were welcomed into many communities who already had remnant British ties, and, further, had been ill-treated by the young American government thus far. Specific to the southeast, families like the McGillivrays, McIntoshs, Colberts, Perrymans, and a host of others had very strong ties to Great Britain already, both economic and familial. Their original Loyalist-progenitor fathers had married women in high-ranking clans with many influential families who then gave birth to mestizo children already of high status who then also were able to create further trade relationships through their command of English. So I would say, at least in the southeast, that it was far less traveling British diplomats that did the trick, but banking on the long-standing ties to British trade goods that had developed among high-ranking, influential native families.

Sources: Kathryn Holland-Braund - Deerskins and Duffels

William Coker and Thomas D. Watson - Indian Traders of the Southeastern Spanish Borderlands

John Alden - John Stuart and the Southern Colonial Frontier

John Caughey - McGillivray of the Creeks

Lots and lots of primary sources (letters of the trade company members, Nicholls and Woodbine correspondence, Spanish Florida governors' correspondence, etc) from University of Florida, University of West Florida, and British Public Record Office.

Edit: Here's a copy of a proclamation that was delivered to many towns in the Creek and Seminole nations in 1814, using Indian messengers to deliver paper correspondence: Imgur Note the specific mentions of Great Britain's willingness to clothe and arm anyone willing to join the British cause. Arms, ammunition, and cloth/clothing were consistently some of the highest-volume, highest-demand items for native customers.

Edit2: I should also mention that most of the messages delivered to these communities, such as the one above, promised to return Indian lands ceded to the United States back to the native people. Whether that would have actually been carried out I am not sure, but the ownership and free use of their land was extremely important to native communities.